Copyright (c) 1999 Association of Mizar Users
environ
vocabulary FUNCT_2, MARGREL1, BVFUNC_1, ZF_LANG, FUNCT_1, RELAT_1, PARTIT1;
notation TARSKI, XBOOLE_0, SUBSET_1, MARGREL1, VALUAT_1, RELAT_1, FUNCT_1,
FRAENKEL, BINARITH, BVFUNC_1;
constructors BINARITH, BVFUNC_1;
clusters MARGREL1, VALUAT_1, BINARITH, FRAENKEL;
requirements SUBSET, BOOLE;
definitions BVFUNC_1;
theorems FUNCT_1, FUNCT_2, MARGREL1, BINARITH, BVFUNC_1, BVFUNC_4, VALUAT_1;
begin
::Chap. 1 Propositional Calculus
reserve Y for non empty set;
theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b) '&' ('not' a 'imp' b) = b
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a 'imp' b) '&' ('not' a 'imp' b),x) = Pj(b,x)
proof
let x be Element of Y;
A2:Pj((a 'imp' b) '&' ('not' a 'imp' b),x)
=Pj(a 'imp' b,x) '&' Pj('not' a 'imp' b,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' Pj('not' a 'imp' b,x) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj('not'
a,x) 'or' Pj(b,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' 'not'
Pj(a,x) 'or' Pj(b,x)) by VALUAT_1:def 5
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' (Pj(a,x) 'or' Pj(b,x)) by MARGREL1:40;
now per cases by MARGREL1:39;
case Pj(a,x)=TRUE;
then Pj((a 'imp' b) '&' ('not' a 'imp' b),x)
=(FALSE 'or' Pj(b,x)) '&' (TRUE 'or' Pj(b,x)) by A2,MARGREL1:41
.=(FALSE 'or' Pj(b,x)) '&' TRUE by BINARITH:19
.=TRUE '&' Pj(b,x) by BINARITH:7
.=Pj(b,x) by MARGREL1:50;
hence thesis;
case Pj(a,x)=FALSE;
then Pj((a 'imp' b) '&' ('not' a 'imp' b),x)
=(TRUE 'or' Pj(b,x)) '&' (FALSE 'or' Pj(b,x)) by A2,MARGREL1:41
.=TRUE '&' (FALSE 'or' Pj(b,x)) by BINARITH:19
.=TRUE '&' Pj(b,x) by BINARITH:7
.=Pj(b,x) by MARGREL1:50;
hence thesis;
end;
hence thesis;
end;
consider k3 being Function such that
A3: ((a 'imp' b) '&' ('not' a 'imp' b))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A4: b=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A3,A4;
hence ((a 'imp' b) '&' ('not' a 'imp' b))=b by A3,A4,FUNCT_1:9;
end;
theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b) '&' (a 'imp' 'not' b) = 'not' a
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a 'imp' b) '&' (a 'imp' 'not' b),x) = Pj('not' a,x)
proof
let x be Element of Y;
A2:Pj((a 'imp' b) '&' (a 'imp' 'not' b),x)
=Pj(a 'imp' b,x) '&' Pj(a 'imp' 'not' b,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' Pj(a 'imp' 'not' b,x) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(a,x) 'or' Pj('not'
b,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(a,x) 'or' 'not'
Pj(b,x)) by VALUAT_1:def 5;
now per cases by MARGREL1:39;
case Pj(b,x)=TRUE;
then Pj((a 'imp' b) '&' (a 'imp' 'not' b),x)
=('not' Pj(a,x) 'or' TRUE) '&' ('not' Pj(a,x) 'or' FALSE) by A2,MARGREL1:41
.=('not' Pj(a,x) 'or' TRUE) '&' 'not' Pj(a,x) by BINARITH:7
.=TRUE '&' 'not' Pj(a,x) by BINARITH:19
.='not' Pj(a,x) by MARGREL1:50
.=Pj('not' a,x) by VALUAT_1:def 5;
hence thesis;
case Pj(b,x)=FALSE;
then Pj((a 'imp' b) '&' (a 'imp' 'not' b),x)
=('not' Pj(a,x) 'or' FALSE) '&' ('not' Pj(a,x) 'or' TRUE) by A2,MARGREL1:41
.='not' Pj(a,x) '&' ('not' Pj(a,x) 'or' TRUE) by BINARITH:7
.=TRUE '&' 'not' Pj(a,x) by BINARITH:19
.='not' Pj(a,x) by MARGREL1:50
.=Pj('not' a,x) by VALUAT_1:def 5;
hence thesis;
end;
hence thesis;
end;
consider k3 being Function such that
A3: ((a 'imp' b) '&' (a 'imp' 'not' b))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A4: 'not' a=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A3,A4;
hence ((a 'imp' b) '&' (a 'imp' 'not' b))='not' a by A3,A4,FUNCT_1:9;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
a 'imp' (b 'or' c) = (a 'imp' b) 'or' (a 'imp' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'imp' (b 'or' c),x) = Pj((a 'imp' b) 'or' (a 'imp' c),x)
proof
let x be Element of Y;
Pj((a 'imp' b) 'or' (a 'imp' c),x)
=Pj(a 'imp' b,x) 'or' Pj(a 'imp' c,x) by BVFUNC_1:def 7
.=('not' Pj(a,x) 'or' Pj(b,x)) 'or' Pj(a 'imp' c,x) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) 'or' ('not'
Pj(a,x) 'or' Pj(c,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' ('not' Pj(a,x) 'or' Pj(b,x))) 'or' Pj(c,x)
by BINARITH:20
.=(('not' Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) 'or' Pj(c,x)
by BINARITH:20
.=('not' Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x) by BINARITH:21
.='not' Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x)) by BINARITH:20
.='not' Pj(a,x) 'or' Pj(b 'or' c,x) by BVFUNC_1:def 7
.=Pj(a 'imp' (b 'or' c),x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'imp' (b 'or' c))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' b) 'or' (a 'imp' c)=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (a 'imp' (b 'or' c))=(a 'imp' b) 'or' (a 'imp' c) by A2,A3,FUNCT_1:9;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
a 'imp' (b '&' c) = (a 'imp' b) '&' (a 'imp' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'imp' (b '&' c),x) = Pj((a 'imp' b) '&' (a 'imp' c),x)
proof
let x be Element of Y;
Pj((a 'imp' b) '&' (a 'imp' c),x)
=Pj(a 'imp' b,x) '&' Pj(a 'imp' c,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' Pj(a 'imp' c,x) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(a,x) 'or' Pj(c,x)) by BVFUNC_1:def 11
.='not' Pj(a,x) 'or' (Pj(b,x) '&' Pj(c,x)) by BINARITH:23
.='not' Pj(a,x) 'or' (Pj(b '&' c,x)) by VALUAT_1:def 6
.=Pj(a 'imp' (b '&' c),x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'imp' (b '&' c))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' b) '&' (a 'imp' c)=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (a 'imp' (b '&' c))=(a 'imp' b) '&' (a 'imp' c) by A2,A3,FUNCT_1:9;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'or' b) 'imp' c = (a 'imp' c) '&' (b 'imp' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a 'or' b) 'imp' c,x) = Pj((a 'imp' c) '&' (b 'imp' c),x)
proof
let x be Element of Y;
Pj((a 'imp' c) '&' (b 'imp' c),x)
=Pj(a 'imp' c,x) '&' Pj(b 'imp' c,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' Pj(c,x)) '&' Pj(b 'imp' c,x) by BVFUNC_1:def 11
.=(Pj(c,x) 'or' 'not' Pj(a,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x)) by BVFUNC_1:def 11
.=Pj(c,x) 'or' ('not' Pj(a,x) '&' 'not' Pj(b,x)) by BINARITH:23
.='not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x) by BINARITH:10
.='not' Pj(a 'or' b,x) 'or' Pj(c,x) by BVFUNC_1:def 7
.=Pj((a 'or' b) 'imp' c,x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: ((a 'or' b) 'imp' c)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' c) '&' (b 'imp' c)=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence ((a 'or' b) 'imp' c)=(a 'imp' c) '&' (b 'imp' c) by A2,A3,FUNCT_1:9;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a '&' b) 'imp' c = (a 'imp' c) 'or' (b 'imp' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a '&' b) 'imp' c,x) = Pj((a 'imp' c) 'or' (b 'imp' c),x)
proof
let x be Element of Y;
Pj((a 'imp' c) 'or' (b 'imp' c),x)
=Pj(a 'imp' c,x) 'or' Pj(b 'imp' c,x) by BVFUNC_1:def 7
.=('not' Pj(a,x) 'or' Pj(c,x)) 'or' Pj(b 'imp' c,x) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(c,x)) 'or' ('not'
Pj(b,x) 'or' Pj(c,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' (Pj(c,x) 'or' 'not' Pj(b,x))) 'or' Pj(c,x)
by BINARITH:20
.=(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x)) 'or' Pj(c,x)
by BINARITH:20
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' (Pj(c,x) 'or' Pj(c,x))
by BINARITH:20
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x) by BINARITH:21
.='not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(c,x) by BINARITH:9
.='not' Pj(a '&' b,x) 'or' Pj(c,x) by VALUAT_1:def 6
.=Pj((a '&' b) 'imp' c,x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: ((a '&' b) 'imp' c)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' c) 'or' (b 'imp' c)=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence ((a '&' b) 'imp' c)=(a 'imp' c) 'or' (b 'imp' c) by A2,A3,FUNCT_1:9;
end;
theorem Th7: for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a '&' b) 'imp' c = a 'imp' (b 'imp' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a '&' b) 'imp' c,x) = Pj(a 'imp' (b 'imp' c),x)
proof
let x be Element of Y;
Pj(a 'imp' (b 'imp' c),x)
='not' Pj(a,x) 'or' Pj(b 'imp' c,x) by BVFUNC_1:def 11
.='not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(c,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x) by BINARITH:20
.='not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(c,x) by BINARITH:9
.='not' Pj(a '&' b,x) 'or' Pj(c,x) by VALUAT_1:def 6
.=Pj((a '&' b) 'imp' c,x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: ((a '&' b) 'imp' c)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a 'imp' (b 'imp' c)=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence ((a '&' b) 'imp' c)=a 'imp' (b 'imp' c) by A2,A3,FUNCT_1:9;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a '&' b) 'imp' c = a 'imp' ('not' b 'or' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a '&' b) 'imp' c,x) = Pj(a 'imp' ('not' b 'or' c),x)
proof
let x be Element of Y;
Pj(a 'imp' ('not' b 'or' c),x)
=Pj(a 'imp' (b 'imp' c),x) by BVFUNC_4:8;
hence thesis by Th7;
end;
consider k3 being Function such that
A2: ((a '&' b) 'imp' c)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a 'imp' ('not' b 'or' c)=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence ((a '&' b) 'imp' c)=a 'imp' ('not' b 'or' c) by A2,A3,FUNCT_1:9;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
a 'imp' (b 'or' c) = (a '&' 'not' b) 'imp' c
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'imp' (b 'or' c),x) = Pj((a '&' 'not' b) 'imp' c,x)
proof
let x be Element of Y;
Pj((a '&' 'not' b) 'imp' c,x)
='not' Pj(a '&' 'not' b,x) 'or' Pj(c,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj('not' b,x)) 'or' Pj(c,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' 'not' Pj('not' b,x)) 'or' Pj(c,x) by BINARITH:9
.=('not' Pj(a,x) 'or' 'not' 'not' Pj(b,x)) 'or' Pj(c,x) by VALUAT_1:def 5
.=('not' Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x) by MARGREL1:40
.='not' Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x)) by BINARITH:20
.='not' Pj(a,x) 'or' Pj(b 'or' c,x) by BVFUNC_1:def 7
.=Pj(a 'imp' (b 'or' c),x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'imp' (b 'or' c))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a '&' 'not' b) 'imp' c =k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (a 'imp' (b 'or' c))=(a '&' 'not' b) 'imp' c by A2,A3,FUNCT_1:9;
end;
theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a '&' (a 'imp' b) = a '&' b
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a '&' (a 'imp' b),x) = Pj(a '&' b,x)
proof
let x be Element of Y;
Pj(a '&' (a 'imp' b),x)
=Pj(a,x) '&' Pj(a 'imp' b,x) by VALUAT_1:def 6
.=Pj(a,x) '&' ('not' Pj(a,x) 'or' Pj(b,x)) by BVFUNC_1:def 11
.=(Pj(a,x) '&' 'not' Pj(a,x)) 'or' (Pj(a,x) '&' Pj(b,x)) by BINARITH:22
.=FALSE 'or' (Pj(a,x) '&' Pj(b,x)) by MARGREL1:46
.=Pj(a,x) '&' Pj(b,x) by BINARITH:7
.=Pj(a '&' b,x) by VALUAT_1:def 6;
hence thesis;
end;
consider k3 being Function such that
A2: (a '&' (a 'imp' b))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a '&' b=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (a '&' (a 'imp' b))=a '&' b by A2,A3,FUNCT_1:9;
end;
theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b) '&' 'not' b = 'not' a '&' 'not' b
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a 'imp' b) '&' 'not' b,x) = Pj('not' a '&' 'not' b,x)
proof
let x be Element of Y;
Pj((a 'imp' b) '&' 'not' b,x)
=Pj(a 'imp' b,x) '&' Pj('not' b,x) by VALUAT_1:def 6
.=Pj('not' b,x) '&' ('not' Pj(a,x) 'or' Pj(b,x)) by BVFUNC_1:def 11
.=(Pj('not' b,x) '&' 'not' Pj(a,x)) 'or' (Pj('not'
b,x) '&' Pj(b,x)) by BINARITH:22
.=(Pj('not' b,x) '&' 'not' Pj(a,x)) 'or' (Pj(b,x) '&' 'not'
Pj(b,x)) by VALUAT_1:def 5
.=(Pj('not' b,x) '&' 'not' Pj(a,x)) 'or' FALSE by MARGREL1:46
.=Pj('not' b,x) '&' 'not' Pj(a,x) by BINARITH:7
.=Pj('not' b,x) '&' Pj('not' a,x) by VALUAT_1:def 5
.=Pj('not' a '&' 'not' b,x) by VALUAT_1:def 6;
hence thesis;
end;
consider k3 being Function such that
A2: ((a 'imp' b) '&' 'not' b)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: 'not' a '&' 'not' b=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence ((a 'imp' b) '&' 'not' b)='not' a '&' 'not' b by A2,A3,FUNCT_1:9;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b) '&' (b 'imp' c) = (a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a 'imp' b) '&' (b 'imp' c),x) =
Pj((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c),x)
proof
let x be Element of Y;
A2:Pj((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c),x)
=Pj((a 'imp' b) '&' (b 'imp' c),x) '&' Pj((a 'imp' c),x) by VALUAT_1:def 6
.=(Pj(a 'imp' b,x) '&' Pj(b 'imp' c,x)) '&' Pj((a 'imp' c),x)
by VALUAT_1:def 6
.=(('not' Pj(a,x) 'or' Pj(b,x)) '&' Pj(b 'imp' c,x)) '&' Pj((a 'imp' c),x)
by BVFUNC_1:def 11
.=(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x))) '&' Pj((a 'imp' c),x)
by BVFUNC_1:def 11
.=(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x)))
'&' ('not' Pj(a,x) 'or' Pj(c,x)) by BVFUNC_1:def 11
.=(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) '&' 'not'
Pj(a,x) 'or'
(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) '&' Pj(c,x)
by BINARITH:22;
A3:Pj((a 'imp' b) '&' (b 'imp' c),x)
=Pj(a 'imp' b,x) '&' Pj(b 'imp' c,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' Pj(b 'imp' c,x)
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))
by BVFUNC_1:def 11;
A4:(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x)))
=(Pj(a 'imp' b,x) '&' ('not' Pj(b,x) 'or' Pj(c,x)))
by BVFUNC_1:def 11
.=Pj(a 'imp' b,x) '&' Pj(b 'imp' c,x)
by BVFUNC_1:def 11
.=Pj((a 'imp' b) '&' (b 'imp' c),x) by VALUAT_1:def 6;
now per cases by MARGREL1:39;
case Pj(a,x)=TRUE & Pj(c,x)=TRUE;
then Pj((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c),x)
=(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x))) '&' FALSE 'or'
(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) '&' TRUE
by A2,MARGREL1:41
.=FALSE 'or' (('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x))) '&' TRUE
by MARGREL1:49
.=FALSE 'or' (('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x)))
by MARGREL1:50
.=Pj((a 'imp' b) '&' (b 'imp' c),x) by A4,BINARITH:7;
hence thesis;
case A5:Pj(a,x)=TRUE & Pj(c,x)=FALSE;
then A6:Pj((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c),x)
=(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x))) '&' FALSE 'or'
(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) '&' FALSE
by A2,MARGREL1:41
.=FALSE '&' (('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x)
'or' Pj(c,x))) by BINARITH:21
.=FALSE by MARGREL1:49;
Pj((a 'imp' b) '&' (b 'imp' c),x)
=(FALSE 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' FALSE) by A3,A5,MARGREL1:41
.=(FALSE 'or' Pj(b,x)) '&' 'not' Pj(b,x) by BINARITH:7
.=Pj(b,x) '&' 'not' Pj(b,x) by BINARITH:7
.=FALSE by MARGREL1:46;
hence thesis by A6;
case Pj(a,x)=FALSE & Pj(c,x)=TRUE;
then Pj((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c),x)
=(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x))) '&' TRUE 'or'
(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) '&' TRUE
by A2,MARGREL1:41
.=TRUE '&' (('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x)))
by BINARITH:21
.=Pj((a 'imp' b) '&' (b 'imp' c),x) by A4,MARGREL1:50;
hence thesis;
case Pj(a,x)=FALSE & Pj(c,x)=FALSE;
then Pj((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c),x)
=TRUE '&' (('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x))) 'or'
FALSE '&' (('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x))) by A2,MARGREL1:41
.=(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) 'or'
FALSE '&' (('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x)))
by MARGREL1:50
.=(('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) 'or' FALSE
by MARGREL1:49
.=Pj((a 'imp' b) '&' (b 'imp' c),x) by A4,BINARITH:7;
hence thesis;
end;
hence thesis;
end;
consider k3 being Function such that
A7: ((a 'imp' b) '&' (b 'imp' c))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A8: (a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)=
k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A7,A8;
hence ((a 'imp' b) '&' (b 'imp' c))=
(a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c) by A7,A8,FUNCT_1:9;
end;
theorem for a being Element of Funcs(Y,BOOLEAN) holds
I_el(Y) 'imp' a = a
proof
let a be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(I_el(Y) 'imp' a,x) = Pj(a,x)
proof
let x be Element of Y;
Pj(I_el(Y) 'imp' a,x)
='not' Pj(I_el(Y),x) 'or' Pj(a,x) by BVFUNC_1:def 11
.='not' TRUE 'or' Pj(a,x) by BVFUNC_1:def 14
.=FALSE 'or' Pj(a,x) by MARGREL1:41;
hence thesis by BINARITH:7;
end;
consider k3 being Function such that
A2: (I_el(Y) 'imp' a)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (I_el(Y) 'imp' a)=a by A2,A3,FUNCT_1:9;
end;
theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'imp' O_el(Y) = 'not' a
proof
let a be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'imp' O_el(Y),x) = Pj('not' a,x)
proof
let x be Element of Y;
Pj(a 'imp' O_el(Y),x)
='not' Pj(a,x) 'or' Pj(O_el(Y),x) by BVFUNC_1:def 11
.='not' Pj(a,x) 'or' FALSE by BVFUNC_1:def 13
.='not' Pj(a,x) by BINARITH:7;
hence thesis by VALUAT_1:def 5;
end;
consider k3 being Function such that
A2: (a 'imp' O_el(Y))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: 'not' a=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (a 'imp' O_el(Y))='not' a by A2,A3,FUNCT_1:9;
end;
theorem for a being Element of Funcs(Y,BOOLEAN) holds
O_el(Y) 'imp' a = I_el(Y)
proof
let a be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(O_el(Y) 'imp' a,x) = TRUE
proof
let x be Element of Y;
Pj(O_el(Y) 'imp' a,x)
='not' Pj(O_el(Y),x) 'or' Pj(a,x) by BVFUNC_1:def 11
.='not' FALSE 'or' Pj(a,x) by BVFUNC_1:def 13
.=TRUE 'or' Pj(a,x) by MARGREL1:41;
hence thesis by BINARITH:19;
end;
hence thesis by BVFUNC_1:def 14;
end;
theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'imp' I_el(Y) = I_el(Y)
proof
let a be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(a 'imp' I_el(Y),x) = TRUE
proof
let x be Element of Y;
Pj(a 'imp' I_el(Y),x)
='not' Pj(a,x) 'or' Pj(I_el(Y),x) by BVFUNC_1:def 11
.='not' Pj(a,x) 'or' TRUE by BVFUNC_1:def 14;
hence thesis by BINARITH:19;
end;
hence thesis by BVFUNC_1:def 14;
end;
theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'imp' 'not' a = 'not' a
proof
let a be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'imp' 'not' a,x) = Pj('not' a,x)
proof
let x be Element of Y;
Pj(a 'imp' 'not' a,x)
='not' Pj(a,x) 'or' Pj('not' a,x) by BVFUNC_1:def 11
.=Pj('not' a,x) 'or' Pj('not' a,x) by VALUAT_1:def 5
.=Pj('not' a,x) by BINARITH:21;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'imp' 'not' a)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: 'not' a=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (a 'imp' 'not' a)='not' a by A2,A3,FUNCT_1:9;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b) '<' (c 'imp' a) 'imp' (c 'imp' b)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume Pj(a 'imp' b,z)=TRUE;
then A1:'not' Pj(a,z) 'or' Pj(b,z) = TRUE by BVFUNC_1:def 11;
A2: Pj(b,z)=TRUE or Pj(b,z)=FALSE by MARGREL1:39;
now per cases by A1,A2,BINARITH:7;
case A3:'not' Pj(a,z)=TRUE;
Pj((c 'imp' a) 'imp' (c 'imp' b),z)
='not' Pj(c 'imp' a,z) 'or' Pj(c 'imp' b,z) by BVFUNC_1:def 11
.='not'( 'not' Pj(c,z) 'or' Pj(a,z)) 'or' Pj(c 'imp' b,z) by BVFUNC_1:def 11
.=('not' 'not' Pj(c,z) '&' TRUE) 'or' Pj(c 'imp' b,z) by A3,BINARITH:10
.=(TRUE '&' Pj(c,z)) 'or' Pj(c 'imp' b,z) by MARGREL1:40
.=Pj(c,z) 'or' Pj(c 'imp' b,z) by MARGREL1:50
.=Pj(c,z) 'or' ('not' Pj(c,z) 'or' Pj(b,z)) by BVFUNC_1:def 11
.=(Pj(c,z) 'or' 'not' Pj(c,z)) 'or' Pj(b,z) by BINARITH:20
.=TRUE 'or' Pj(b,z) by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
case A4:Pj(b,z)=TRUE;
Pj((c 'imp' a) 'imp' (c 'imp' b),z)
='not' Pj(c 'imp' a,z) 'or' Pj(c 'imp' b,z) by BVFUNC_1:def 11
.='not'( 'not' Pj(c,z) 'or' Pj(a,z)) 'or' Pj(c 'imp' b,z) by BVFUNC_1:def 11
.='not'( 'not' Pj(c,z) 'or' Pj(a,z)) 'or' ('not'
Pj(c,z) 'or' TRUE) by A4,BVFUNC_1:def 11
.='not'( 'not' Pj(c,z) 'or' Pj(a,z)) 'or' TRUE by BINARITH:19
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'eqv' b) '<' (a 'eqv' c) 'eqv' (b 'eqv' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a 'eqv' b,z)=TRUE;
Pj(a 'eqv' b,z)
=Pj((a 'imp' b) '&' (b 'imp' a),z) by BVFUNC_4:7
.=Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z) by VALUAT_1:def 6;
then A2:Pj(a 'imp' b,z)=TRUE & Pj(b 'imp' a,z)=TRUE by A1,MARGREL1:45;
then A3:'not' Pj(a,z) 'or' Pj(b,z) = TRUE by BVFUNC_1:def 11;
A4: Pj(b,z)=TRUE or Pj(b,z)=FALSE by MARGREL1:39;
A5: Pj(b 'imp' a,z) = 'not' Pj(b,z) 'or' Pj(a,z) by BVFUNC_1:def 11;
Pj(a,z)=TRUE or Pj(a,z)=FALSE by MARGREL1:39;
then A6:'not' Pj(b,z)=TRUE or Pj(a,z)=TRUE by A2,A5,BINARITH:7;
A7:Pj((a 'eqv' c) 'eqv' (b 'eqv' c),z)
=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(a,z))) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(b,z))) 'or'
(('not' Pj(a,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(a,z))))
proof
Pj((a 'eqv' c) 'eqv' (b 'eqv' c),z)
=Pj(((a 'eqv' c) 'imp' (b 'eqv' c)) '&' ((b 'eqv' c) 'imp' (a 'eqv' c)),z)
by BVFUNC_4:7
.=Pj((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'eqv' c)) '&'
((b 'eqv' c) 'imp' (a 'eqv' c)),z)
by BVFUNC_4:7
.=Pj((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'imp' c) '&' (c 'imp' b)) '&'
((b 'eqv' c) 'imp' (a 'eqv' c)),z)
by BVFUNC_4:7
.=Pj((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'imp' c) '&' (c 'imp' b)) '&'
(((b 'imp' c) '&' (c 'imp' b)) 'imp' (a 'eqv' c)),z)
by BVFUNC_4:7
.=Pj((((a 'imp' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&'
(((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a))),z)
by BVFUNC_4:7
.=Pj(((('not'
a 'or' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&'
(((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a))),z)
by BVFUNC_4:8
.=Pj(((('not' a 'or' c) '&' ('not'
c 'or' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&'
(((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a))),z)
by BVFUNC_4:8
.=Pj(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not'
b 'or' c) '&' (c 'imp' b))) '&'
(((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a))),z)
by BVFUNC_4:8
.=Pj(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp'
(('not' b 'or' c) '&' ('not'
c 'or' b))) '&'
(((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a))),z)
by BVFUNC_4:8
.=Pj(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&' ((('not'
b 'or' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a))),z)
by BVFUNC_4:8
.=Pj(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
((('not' b 'or' c) '&' ('not'
c 'or' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a))),z) by BVFUNC_4:8
.=Pj(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' (('not'
a 'or' c) '&' (c 'imp' a))),z) by BVFUNC_4:8
.=Pj(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_4:8
.=Pj(('not'( ('not' a 'or' c) '&' ('not' c 'or' a))
'or' (('not' b 'or' c) '&' ('not' c 'or' b))) '&'
((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_4:8
.=Pj(('not'( ('not' a 'or' c) '&' ('not' c 'or' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
('not'( ('not' b 'or' c) '&' ('not' c 'or' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_4:8
.=Pj((('not'( 'not' a 'or' c) 'or' 'not'( 'not' c 'or' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
('not'( ('not' b 'or' c) '&' ('not' c 'or' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:17
.=Pj((('not'( 'not' a 'or' c) 'or' 'not'( 'not' c 'or' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
(('not'( 'not' b 'or' c) 'or' 'not'( 'not' c 'or' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:17
.=Pj(((('not' 'not' a '&' 'not' c) 'or' 'not'( 'not' c 'or' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
(('not'( 'not' b 'or' c) 'or' 'not'( 'not' c 'or' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:16
.=Pj(((('not' 'not' a '&' 'not' c) 'or' ('not' 'not' c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
(('not'( 'not' b 'or' c) 'or' 'not'( 'not' c 'or' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:16
.=Pj(((('not' 'not' a '&' 'not' c) 'or' ('not' 'not' c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
((('not' 'not' b '&' 'not' c) 'or' 'not'( 'not' c 'or' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:16
.=Pj(((('not' 'not' a '&' 'not' c) 'or' ('not' 'not' c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
((('not' 'not' b '&' 'not' c) 'or' ('not' 'not' c '&' 'not' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:16
.=Pj((((a '&' 'not' c) 'or' ('not' 'not' c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
((('not' 'not' b '&' 'not' c) 'or' ('not' 'not' c '&' 'not' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:4
.=Pj((((a '&' 'not' c) 'or' (c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
((('not' 'not' b '&' 'not' c) 'or' ('not' 'not' c '&' 'not' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:4
.=Pj((((a '&' 'not' c) 'or' (c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
(((b '&' 'not' c) 'or' ('not' 'not' c '&' 'not' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:4
.=Pj((((a '&' 'not' c) 'or' (c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&'
(((b '&' 'not' c) 'or' (c '&' 'not' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a))),z)
by BVFUNC_1:4
.=Pj(((a '&' 'not' c) 'or' (c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b)),z) '&'
Pj(((b '&' 'not' c) 'or' (c '&' 'not' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a)),z)
by VALUAT_1:def 6
.=(Pj((a '&' 'not' c) 'or' (c '&' 'not' a),z) 'or'
Pj(('not' b 'or' c) '&' ('not' c 'or' b),z)) '&'
Pj(((b '&' 'not' c) 'or' (c '&' 'not' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a)),z)
by BVFUNC_1:def 7
.=(Pj((a '&' 'not' c) 'or' (c '&' 'not' a),z) 'or'
Pj(('not' b 'or' c) '&' ('not' c 'or' b),z)) '&'
(Pj((b '&' 'not' c) 'or' (c '&' 'not' b),z) 'or'
Pj(('not' a 'or' c) '&' ('not' c 'or' a),z))
by BVFUNC_1:def 7
.=((Pj(a '&' 'not' c,z) 'or' Pj(c '&' 'not' a,z)) 'or'
Pj(('not' b 'or' c) '&' ('not' c 'or' b),z)) '&'
(Pj((b '&' 'not' c) 'or' (c '&' 'not' b),z) 'or'
Pj(('not' a 'or' c) '&' ('not' c 'or' a),z))
by BVFUNC_1:def 7
.=((Pj(a '&' 'not' c,z) 'or' Pj(c '&' 'not' a,z)) 'or'
(Pj('not' b 'or' c,z) '&' Pj('not' c 'or' b,z))) '&'
(Pj((b '&' 'not' c) 'or' (c '&' 'not' b),z) 'or'
Pj(('not' a 'or' c) '&' ('not' c 'or' a),z))
by VALUAT_1:def 6
.=((Pj(a '&' 'not' c,z) 'or' Pj(c '&' 'not' a,z)) 'or'
(Pj('not' b 'or' c,z) '&' Pj('not' c 'or' b,z))) '&'
((Pj(b '&' 'not' c,z) 'or' Pj(c '&' 'not' b,z)) 'or'
Pj(('not' a 'or' c) '&' ('not' c 'or' a),z))
by BVFUNC_1:def 7
.=((Pj(a '&' 'not' c,z) 'or' Pj(c '&' 'not' a,z)) 'or'
(Pj('not' b 'or' c,z) '&' Pj('not' c 'or' b,z))) '&'
((Pj(b '&' 'not' c,z) 'or' Pj(c '&' 'not' b,z)) 'or'
(Pj('not' a 'or' c,z) '&' Pj('not' c 'or' a,z)))
by VALUAT_1:def 6
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' Pj(c '&' 'not' a,z)) 'or'
(Pj('not' b 'or' c,z) '&' Pj('not' c 'or' b,z))) '&'
((Pj(b '&' 'not' c,z) 'or' Pj(c '&' 'not' b,z)) 'or'
(Pj('not' a 'or' c,z) '&' Pj('not' c 'or' a,z)))
by VALUAT_1:def 6
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' a,z))) 'or'
(Pj('not' b 'or' c,z) '&' Pj('not' c 'or' b,z))) '&'
((Pj(b '&' 'not' c,z) 'or' Pj(c '&' 'not' b,z)) 'or'
(Pj('not' a 'or' c,z) '&' Pj('not' c 'or' a,z)))
by VALUAT_1:def 6
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' a,z))) 'or'
((Pj('not' b,z) 'or' Pj(c,z)) '&' Pj('not' c 'or' b,z))) '&'
((Pj(b '&' 'not' c,z) 'or' Pj(c '&' 'not' b,z)) 'or'
(Pj('not' a 'or' c,z) '&' Pj('not' c 'or' a,z)))
by BVFUNC_1:def 7
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' a,z))) 'or'
((Pj('not' b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
((Pj(b '&' 'not' c,z) 'or' Pj(c '&' 'not' b,z)) 'or'
(Pj('not' a 'or' c,z) '&' Pj('not' c 'or' a,z)))
by BVFUNC_1:def 7
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' a,z))) 'or'
((Pj('not' b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' Pj(c '&' 'not' b,z)) 'or'
(Pj('not' a 'or' c,z) '&' Pj('not' c 'or' a,z)))
by VALUAT_1:def 6
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' a,z))) 'or'
((Pj('not' b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' b,z))) 'or'
(Pj('not' a 'or' c,z) '&' Pj('not' c 'or' a,z)))
by VALUAT_1:def 6
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' a,z))) 'or'
((Pj('not' b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' b,z))) 'or'
((Pj('not' a,z) 'or' Pj(c,z)) '&' Pj('not' c 'or' a,z)))
by BVFUNC_1:def 7
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' a,z))) 'or'
((Pj('not' b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' b,z))) 'or'
((Pj('not' a,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(a,z))))
by BVFUNC_1:def 7
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(a,z))) 'or'
((Pj('not' b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' b,z))) 'or'
((Pj('not' a,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(a,z))))
by VALUAT_1:def 5
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(a,z))) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' Pj('not' b,z))) 'or'
((Pj('not' a,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(a,z))))
by VALUAT_1:def 5
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(a,z))) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(b,z))) 'or'
((Pj('not' a,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(a,z))))
by VALUAT_1:def 5
.=(((Pj(a,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(a,z))) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(b,z))) 'or'
(('not' Pj(a,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(a,z))))
by VALUAT_1:def 5;
hence thesis;
end;
now per cases by A3,A4,BINARITH:7;
case A8:'not' Pj(a,z)=TRUE;
then A9:Pj(a,z)=FALSE by MARGREL1:41;
then Pj((a 'eqv' c) 'eqv' (b 'eqv' c),z)
=((FALSE 'or' (Pj(c,z) '&' TRUE)) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(b,z))) 'or'
((TRUE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' FALSE)))
by A7,A8,MARGREL1:49
.=((FALSE 'or' Pj(c,z)) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(b,z))) 'or'
((TRUE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' FALSE)))
by MARGREL1:50
.=(Pj(c,z) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(b,z))) 'or'
((TRUE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' FALSE)))
by BINARITH:7
.=(Pj(c,z) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(b,z))) 'or'
((TRUE 'or' Pj(c,z)) '&' Pj('not' c,z)))
by BINARITH:7
.=(Pj(c,z) 'or'
(('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not' Pj(b,z))) 'or'
(TRUE '&' Pj('not' c,z)))
by BINARITH:19
.=(Pj(c,z) 'or' (('not' Pj(b,z) 'or' Pj(c,z)) '&' (Pj('not'
c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not'
Pj(b,z))) 'or' Pj('not' c,z))
by MARGREL1:50
.=((Pj(c,z) 'or' (Pj(c,z) 'or' 'not' Pj(b,z))) '&'
(Pj(c,z) 'or' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not'
Pj(b,z))) 'or' Pj('not' c,z)) by BINARITH:23
.=(((Pj(c,z) 'or' Pj(c,z)) 'or' 'not' Pj(b,z)) '&'
(Pj(c,z) 'or' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not'
Pj(b,z))) 'or' Pj('not' c,z)) by BINARITH:20
.=((Pj(c,z) 'or' 'not' Pj(b,z)) '&'
(Pj(c,z) 'or' (Pj('not' c,z) 'or' Pj(b,z)))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not'
Pj(b,z))) 'or' Pj('not' c,z))
by BINARITH:21
.=((Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj(c,z) 'or' Pj('not' c,z)) 'or' Pj(b,z))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not'
Pj(b,z))) 'or' Pj('not' c,z)) by BINARITH:20
.=((Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj(c,z) 'or' 'not' Pj(c,z)) 'or' Pj(b,z))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not'
Pj(b,z))) 'or' Pj('not' c,z)) by VALUAT_1:def 5
.=((Pj(c,z) 'or' 'not' Pj(b,z)) '&'
(TRUE 'or' Pj(b,z))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not'
Pj(b,z))) 'or' Pj('not' c,z)) by BINARITH:18
.=(TRUE '&' (Pj(c,z) 'or' 'not' Pj(b,z))) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' 'not'
Pj(b,z))) 'or' Pj('not' c,z)) by BINARITH:19
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
(((Pj(b,z) '&' Pj('not' c,z)) 'or'
(Pj(c,z) '&' 'not' Pj(b,z))) 'or' Pj('not' c,z))
by MARGREL1:50
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj(b,z) '&' Pj('not' c,z)) 'or' (Pj('not' c,z) 'or'
(Pj(c,z) '&' 'not' Pj(b,z)))) by BINARITH:20
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj(b,z) '&' Pj('not' c,z)) 'or'
((Pj('not' c,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' 'not' Pj(b,z))))
by BINARITH:23
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj(b,z) '&' Pj('not' c,z)) 'or'
(('not' Pj(c,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' 'not' Pj(b,z))))
by VALUAT_1:def 5
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj(b,z) '&' Pj('not' c,z)) 'or'
(TRUE '&' (Pj('not' c,z) 'or' 'not' Pj(b,z))))
by BINARITH:18
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj('not' c,z) 'or' 'not' Pj(b,z)) 'or' (Pj(b,z) '&' Pj('not'
c,z))) by MARGREL1:50
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
(((Pj('not' c,z) 'or' 'not' Pj(b,z)) 'or' Pj(b,z)) '&'
((Pj('not' c,z) 'or' 'not' Pj(b,z)) 'or' Pj('not' c,z)))
by BINARITH:23
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj('not' c,z) 'or' ('not' Pj(b,z) 'or' Pj(b,z))) '&'
((Pj('not' c,z) 'or' 'not' Pj(b,z)) 'or' Pj('not' c,z)))
by BINARITH:20
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
((Pj('not' c,z) 'or' TRUE) '&'
((Pj('not' c,z) 'or' 'not' Pj(b,z)) 'or' Pj('not' c,z)))
by BINARITH:18
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
(TRUE '&' ((Pj('not' c,z) 'or' 'not' Pj(b,z)) 'or' Pj('not' c,z)))
by BINARITH:19
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
(('not' Pj(b,z) 'or' Pj('not' c,z)) 'or' Pj('not' c,z)) by MARGREL1:50
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&'
('not' Pj(b,z) 'or' (Pj('not' c,z) 'or' Pj('not' c,z)))
by BINARITH:20
.=(Pj(c,z) 'or' 'not' Pj(b,z)) '&' ('not' Pj(b,z) 'or' Pj('not' c,z))
by BINARITH:21
.=('not' Pj(b,z) '&' (Pj(c,z) 'or' 'not' Pj(b,z))) 'or'
((Pj(c,z) 'or' 'not' Pj(b,z)) '&' Pj('not' c,z)) by BINARITH:22
.=(('not' Pj(b,z) '&' Pj(c,z)) 'or' ('not' Pj(b,z) '&' 'not' Pj(b,z))) 'or'
(Pj('not' c,z) '&' (Pj(c,z) 'or' 'not' Pj(b,z)))
by BINARITH:22
.=(('not' Pj(b,z) '&' Pj(c,z)) 'or' 'not' Pj(b,z)) 'or'
(Pj('not' c,z) '&' (Pj(c,z) 'or' 'not' Pj(b,z)))
by BINARITH:16
.=(('not' Pj(b,z) '&' Pj(c,z)) 'or' 'not' Pj(b,z)) 'or'
(Pj('not' c,z) '&' Pj(c,z) 'or' Pj('not' c,z) '&' 'not' Pj(b,z))
by BINARITH:22
.=(('not' Pj(b,z) '&' Pj(c,z)) 'or' 'not' Pj(b,z)) 'or'
((Pj(c,z) '&' 'not' Pj(c,z)) 'or' (Pj('not' c,z) '&' 'not'
Pj(b,z))) by VALUAT_1:def 5
.=(('not' Pj(b,z) '&' Pj(c,z)) 'or' 'not' Pj(b,z)) 'or'
(FALSE 'or' (Pj('not' c,z) '&' 'not' Pj(b,z)))
by MARGREL1:46
.=('not' Pj(b,z) 'or' ('not' Pj(b,z) '&' Pj(c,z))) 'or'
(Pj('not' c,z) '&' 'not' Pj(b,z)) by BINARITH:7
.='not' Pj(b,z) 'or' (('not' Pj(b,z) '&' Pj(c,z)) 'or'
(Pj('not' c,z) '&' 'not' Pj(b,z)))
by BINARITH:20
.=TRUE by A6,A9,BINARITH:19,MARGREL1:43;
hence thesis;
case A10:Pj(b,z)=TRUE;
then 'not' Pj(b,z)=FALSE by MARGREL1:41;
then Pj((a 'eqv' c) 'eqv' (b 'eqv' c),z)
=((Pj('not' c,z) 'or' (Pj(c,z) '&' FALSE)) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE))) '&'
(((TRUE '&' Pj('not' c,z)) 'or' (Pj(c,z) '&' FALSE)) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE)))
by A6,A7,A10,MARGREL1:43,50
.=((Pj('not' c,z) 'or' (FALSE '&' Pj(c,z))) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE))) '&'
((Pj('not' c,z) 'or' (Pj(c,z) '&' FALSE)) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE))) by MARGREL1:50
.=((Pj('not' c,z) 'or' FALSE) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE))) '&'
((Pj('not' c,z) 'or' (FALSE '&' Pj(c,z))) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE)))
by MARGREL1:49
.=((Pj('not' c,z) 'or' FALSE) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE))) '&'
((Pj('not' c,z) 'or' FALSE) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE)))
by MARGREL1:49
.=((Pj('not' c,z) 'or' FALSE) 'or'
((FALSE 'or' Pj(c,z)) '&' TRUE)) '&'
((Pj('not' c,z) 'or' FALSE) 'or'
((FALSE 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' TRUE)))
by BINARITH:19
.=((Pj('not' c,z) 'or' FALSE) 'or'
((FALSE 'or' Pj(c,z)) '&' TRUE)) '&'
((Pj('not' c,z) 'or' FALSE) 'or'
((FALSE 'or' Pj(c,z)) '&' TRUE))
by BINARITH:19
.=((Pj('not' c,z) 'or' FALSE) 'or'
(Pj(c,z) '&' TRUE)) '&'
((Pj('not' c,z) 'or' FALSE) 'or'
((FALSE 'or' Pj(c,z)) '&' TRUE))
by BINARITH:7
.=((Pj('not' c,z) 'or' FALSE) 'or'
(Pj(c,z) '&' TRUE)) '&'
((Pj('not' c,z) 'or' FALSE) 'or'
(Pj(c,z) '&' TRUE))
by BINARITH:7
.=(Pj('not' c,z) 'or'
(Pj(c,z) '&' TRUE)) '&'
((Pj('not' c,z) 'or' FALSE) 'or'
(Pj(c,z) '&' TRUE))
by BINARITH:7
.=(Pj('not' c,z) 'or' (TRUE '&' Pj(c,z))) '&'
(Pj('not' c,z) 'or' (Pj(c,z) '&' TRUE)) by BINARITH:7
.=(Pj('not' c,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' (TRUE '&' Pj(c,z)))
by MARGREL1:50
.=(Pj('not' c,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(c,z))
by MARGREL1:50
.=('not' Pj(c,z) 'or' Pj(c,z)) '&' (Pj('not' c,z) 'or' Pj(c,z))
by VALUAT_1:def 5
.=('not' Pj(c,z) 'or' Pj(c,z)) '&' ('not' Pj(c,z) 'or' Pj(c,z))
by VALUAT_1:def 5
.=TRUE '&' ('not' Pj(c,z) 'or' Pj(c,z)) by BINARITH:18
.=TRUE '&' TRUE by BINARITH:18
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'eqv' b) '<' (a 'imp' c) 'eqv' (b 'imp' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a 'eqv' b,z)=TRUE;
Pj(a 'eqv' b,z)
=Pj((a 'imp' b) '&' (b 'imp' a),z) by BVFUNC_4:7
.=Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z) by VALUAT_1:def 6;
then A2:Pj(a 'imp' b,z)=TRUE & Pj(b 'imp' a,z)=TRUE by A1,MARGREL1:45;
then A3:'not' Pj(a,z) 'or' Pj(b,z) = TRUE by BVFUNC_1:def 11;
A4: Pj(b,z)=TRUE or Pj(b,z)=FALSE by MARGREL1:39;
A5: Pj(b 'imp' a,z) = 'not' Pj(b,z) 'or' Pj(a,z) by BVFUNC_1:def 11;
A6:'not' Pj(b,z) 'or' Pj(a,z) = TRUE by A2,BVFUNC_1:def 11;
Pj(a,z)=TRUE or Pj(a,z)=FALSE by MARGREL1:39;
then A7:'not' Pj(b,z)=TRUE or Pj(a,z)=TRUE by A2,A5,BINARITH:7;
A8:Pj((a 'imp' c) 'eqv' (b 'imp' c),z)
=Pj(((a 'imp' c) 'imp' (b 'imp' c)) '&' ((b 'imp' c) 'imp' (a 'imp' c)),z)
by BVFUNC_4:7
.=Pj((a 'imp' c) 'imp' (b 'imp' c),z) '&'
Pj((b 'imp' c) 'imp' (a 'imp' c),z)
by VALUAT_1:def 6
.=('not' Pj(a 'imp' c,z) 'or' Pj(b 'imp' c,z)) '&'
Pj((b 'imp' c) 'imp' (a 'imp' c),z)
by BVFUNC_1:def 11
.=('not' Pj(a 'imp' c,z) 'or' Pj(b 'imp' c,z)) '&'
('not' Pj(b 'imp' c,z) 'or' Pj(a 'imp' c,z))
by BVFUNC_1:def 11
.=('not'( 'not' Pj(a,z) 'or' Pj(c,z)) 'or' Pj(b 'imp' c,z)) '&'
('not' Pj(b 'imp' c,z) 'or' Pj(a 'imp' c,z))
by BVFUNC_1:def 11
.=('not'( 'not' Pj(a,z) 'or' Pj(c,z)) 'or' ('not' Pj(b,z) 'or' Pj(c,z))) '&'
('not' Pj(b 'imp' c,z) 'or' Pj(a 'imp' c,z))
by BVFUNC_1:def 11
.=('not'( 'not' Pj(a,z) 'or' Pj(c,z)) 'or' ('not' Pj(b,z) 'or' Pj(c,z))) '&'
('not'( 'not' Pj(b,z) 'or' Pj(c,z)) 'or' Pj(a 'imp' c,z))
by BVFUNC_1:def 11
.=('not'( 'not' Pj(a,z) 'or' Pj(c,z)) 'or' ('not' Pj(b,z) 'or' Pj(c,z))) '&'
('not'( 'not' Pj(b,z) 'or' Pj(c,z)) 'or' ('not' Pj(a,z) 'or' Pj(c,z)))
by BVFUNC_1:def 11
.=(('not' 'not' Pj(a,z) '&' 'not' Pj(c,z)) 'or' ('not' Pj(b,z)
'or' Pj(c,z))) '&'
('not'( 'not' Pj(b,z) 'or' Pj(c,z)) 'or' ('not' Pj(a,z) 'or' Pj(c,z)))
by BINARITH:10
.=(('not' 'not' Pj(a,z) '&' 'not' Pj(c,z)) 'or' ('not'
Pj(b,z) 'or' Pj(c,z))) '&'
(('not' 'not' Pj(b,z) '&' 'not' Pj(c,z)) 'or' ('not' Pj(a,z) 'or' Pj(c,z)))
by BINARITH:10
.=((Pj(a,z) '&' 'not' Pj(c,z)) 'or' ('not' Pj(b,z) 'or' Pj(c,z))) '&'
(('not' 'not' Pj(b,z) '&' 'not' Pj(c,z)) 'or' ('not' Pj(a,z) 'or' Pj(c,z)))
by MARGREL1:40
.=((Pj(a,z) '&' 'not' Pj(c,z)) 'or' ('not' Pj(b,z) 'or' Pj(c,z))) '&'
((Pj(b,z) '&' 'not' Pj(c,z)) 'or' ('not' Pj(a,z) 'or' Pj(c,z)))
by MARGREL1:40;
now per cases by A3,A4,BINARITH:7;
case A9:'not' Pj(a,z)=TRUE;
then A10:Pj(a,z)=FALSE by MARGREL1:41;
then 'not' Pj(b,z)=TRUE by A6,BINARITH:7;
then Pj(b,z)=FALSE by MARGREL1:41;
then Pj((a 'imp' c) 'eqv' (b 'imp' c),z)
=(FALSE 'or' (TRUE 'or' Pj(c,z))) '&'
((FALSE '&' 'not'
Pj(c,z)) 'or' (TRUE 'or' Pj(c,z))) by A8,A9,A10,MARGREL1:49
.=(FALSE 'or' (TRUE 'or' Pj(c,z))) '&'
(FALSE 'or' (TRUE 'or' Pj(c,z))) by MARGREL1:49
.=(TRUE 'or' Pj(c,z)) '&' (FALSE 'or' (TRUE 'or' Pj(c,z))) by BINARITH:7
.=(TRUE 'or' Pj(c,z)) '&' (TRUE 'or' Pj(c,z)) by BINARITH:7
.=TRUE '&' (TRUE 'or' Pj(c,z)) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
case A11:Pj(b,z)=TRUE;
then 'not' Pj(b,z)=FALSE by MARGREL1:41;
then Pj((a 'imp' c) 'eqv' (b 'imp' c),z)
=('not' Pj(c,z) 'or' (FALSE 'or' Pj(c,z))) '&'
((TRUE '&' 'not' Pj(c,z)) 'or' (FALSE 'or' Pj(c,z)))
by A7,A8,A11,MARGREL1:43,50
.=('not' Pj(c,z) 'or' (FALSE 'or' Pj(c,z))) '&'
('not' Pj(c,z) 'or' (FALSE 'or' Pj(c,z)))
by MARGREL1:50
.=('not' Pj(c,z) 'or' Pj(c,z)) '&'
('not' Pj(c,z) 'or' (FALSE 'or' Pj(c,z)))
by BINARITH:7
.=('not' Pj(c,z) 'or' Pj(c,z)) '&' ('not' Pj(c,z) 'or' Pj(c,z))
by BINARITH:7
.=TRUE '&' ('not' Pj(c,z) 'or' Pj(c,z)) by BINARITH:18
.=TRUE '&' TRUE by BINARITH:18
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'eqv' b) '<' (c 'imp' a) 'eqv' (c 'imp' b)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a 'eqv' b,z)=TRUE;
Pj(a 'eqv' b,z)
=Pj((a 'imp' b) '&' (b 'imp' a),z) by BVFUNC_4:7
.=Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z) by VALUAT_1:def 6;
then A2:Pj(a 'imp' b,z)=TRUE & Pj(b 'imp' a,z)=TRUE by A1,MARGREL1:45;
then A3:'not' Pj(a,z) 'or' Pj(b,z) = TRUE by BVFUNC_1:def 11;
A4: Pj(b,z)=TRUE or Pj(b,z)=FALSE by MARGREL1:39;
A5: Pj(b 'imp' a,z) = 'not' Pj(b,z) 'or' Pj(a,z) by BVFUNC_1:def 11;
A6:'not' Pj(b,z) 'or' Pj(a,z) = TRUE by A2,BVFUNC_1:def 11;
Pj(a,z)=TRUE or Pj(a,z)=FALSE by MARGREL1:39;
then A7:'not' Pj(b,z)=TRUE or Pj(a,z)=TRUE by A2,A5,BINARITH:7;
A8:Pj((c 'imp' a) 'eqv' (c 'imp' b),z)
=Pj(((c 'imp' a) 'imp' (c 'imp' b)) '&' ((c 'imp' b) 'imp' (c 'imp' a)),z)
by BVFUNC_4:7
.=Pj((c 'imp' a) 'imp' (c 'imp' b),z) '&'
Pj((c 'imp' b) 'imp' (c 'imp' a),z)
by VALUAT_1:def 6
.=('not' Pj(c 'imp' a,z) 'or' Pj(c 'imp' b,z)) '&'
Pj((c 'imp' b) 'imp' (c 'imp' a),z)
by BVFUNC_1:def 11
.=('not' Pj(c 'imp' a,z) 'or' Pj(c 'imp' b,z)) '&'
('not' Pj(c 'imp' b,z) 'or' Pj(c 'imp' a,z))
by BVFUNC_1:def 11
.=('not'( 'not' Pj(c,z) 'or' Pj(a,z)) 'or' Pj(c 'imp' b,z)) '&'
('not' Pj(c 'imp' b,z) 'or' Pj(c 'imp' a,z))
by BVFUNC_1:def 11
.=('not'( 'not' Pj(c,z) 'or' Pj(a,z)) 'or' ('not' Pj(c,z) 'or' Pj(b,z))) '&'
('not' Pj(c 'imp' b,z) 'or' Pj(c 'imp' a,z))
by BVFUNC_1:def 11
.=('not'( 'not' Pj(c,z) 'or' Pj(a,z)) 'or' ('not' Pj(c,z) 'or' Pj(b,z))) '&'
('not'( 'not' Pj(c,z) 'or' Pj(b,z)) 'or' Pj(c 'imp' a,z))
by BVFUNC_1:def 11
.=('not'( 'not' Pj(c,z) 'or' Pj(a,z)) 'or' ('not' Pj(c,z) 'or' Pj(b,z))) '&'
('not'( 'not' Pj(c,z) 'or' Pj(b,z)) 'or' ('not' Pj(c,z) 'or' Pj(a,z)))
by BVFUNC_1:def 11
.=(('not' 'not' Pj(c,z) '&' 'not' Pj(a,z)) 'or' ('not'
Pj(c,z) 'or' Pj(b,z))) '&'
('not'( 'not' Pj(c,z) 'or' Pj(b,z)) 'or' ('not' Pj(c,z) 'or' Pj(a,z)))
by BINARITH:10
.=(('not' 'not' Pj(c,z) '&' 'not' Pj(a,z)) 'or' ('not'
Pj(c,z) 'or' Pj(b,z))) '&'
(('not' 'not' Pj(c,z) '&' 'not' Pj(b,z)) 'or' ('not' Pj(c,z) 'or' Pj(a,z)))
by BINARITH:10
.=((Pj(c,z) '&' 'not' Pj(a,z)) 'or' ('not' Pj(c,z) 'or' Pj(b,z))) '&'
(('not' 'not' Pj(c,z) '&' 'not' Pj(b,z)) 'or' ('not' Pj(c,z) 'or' Pj(a,z)))
by MARGREL1:40
.=((Pj(c,z) '&' 'not' Pj(a,z)) 'or' ('not' Pj(c,z) 'or' Pj(b,z))) '&'
((Pj(c,z) '&' 'not' Pj(b,z)) 'or' ('not' Pj(c,z) 'or' Pj(a,z)))
by MARGREL1:40;
now per cases by A3,A4,BINARITH:7;
case A9:'not' Pj(a,z)=TRUE;
then A10:Pj(a,z)=FALSE by MARGREL1:41;
then 'not' Pj(b,z)=TRUE by A6,BINARITH:7;
then Pj(b,z)=FALSE by MARGREL1:41;
then Pj((c 'imp' a) 'eqv' (c 'imp' b),z)
=(Pj(c,z) 'or' ('not' Pj(c,z) 'or' FALSE)) '&'
((TRUE '&' Pj(c,z)) 'or' ('not' Pj(c,z) 'or' FALSE))
by A8,A9,A10,MARGREL1:50
.=(Pj(c,z) 'or' ('not' Pj(c,z) 'or' FALSE)) '&'
(Pj(c,z) 'or' ('not' Pj(c,z) 'or' FALSE))
by MARGREL1:50
.=(Pj(c,z) 'or' 'not' Pj(c,z)) '&'
(Pj(c,z) 'or' ('not' Pj(c,z) 'or' FALSE))
by BINARITH:7
.=(Pj(c,z) 'or' 'not' Pj(c,z)) '&' (Pj(c,z) 'or' 'not' Pj(c,z))
by BINARITH:7
.=TRUE '&' (Pj(c,z) 'or' 'not' Pj(c,z))
by BINARITH:18
.=TRUE '&' TRUE by BINARITH:18
.=TRUE by MARGREL1:45;
hence thesis;
case A11:Pj(b,z)=TRUE;
then 'not' Pj(b,z)=FALSE by MARGREL1:41;
then Pj((c 'imp' a) 'eqv' (c 'imp' b),z)
=(FALSE 'or' ('not' Pj(c,z) 'or' TRUE)) '&'
((FALSE '&' Pj(c,z)) 'or' ('not' Pj(c,z) 'or' TRUE))
by A7,A8,A11,MARGREL1:43,49
.=(FALSE 'or' ('not' Pj(c,z) 'or' TRUE)) '&'
(FALSE 'or' ('not' Pj(c,z) 'or' TRUE))
by MARGREL1:49
.=('not' Pj(c,z) 'or' TRUE) '&'
(FALSE 'or' ('not' Pj(c,z) 'or' TRUE))
by BINARITH:7
.=('not' Pj(c,z) 'or' TRUE) '&' ('not' Pj(c,z) 'or' TRUE)
by BINARITH:7
.=TRUE '&' ('not' Pj(c,z) 'or' TRUE)
by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'eqv' b) '<' (a '&' c) 'eqv' (b '&' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a 'eqv' b,z)=TRUE;
Pj(a 'eqv' b,z)
=Pj((a 'imp' b) '&' (b 'imp' a),z) by BVFUNC_4:7
.=Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z) by VALUAT_1:def 6;
then A2:Pj(a 'imp' b,z)=TRUE & Pj(b 'imp' a,z)=TRUE by A1,MARGREL1:45;
then A3:'not' Pj(a,z) 'or' Pj(b,z) = TRUE by BVFUNC_1:def 11;
A4: Pj(b,z)=TRUE or Pj(b,z)=FALSE by MARGREL1:39;
A5: Pj(b 'imp' a,z) = 'not' Pj(b,z) 'or' Pj(a,z) by BVFUNC_1:def 11;
A6:'not' Pj(b,z) 'or' Pj(a,z) = TRUE by A2,BVFUNC_1:def 11;
Pj(a,z)=TRUE or Pj(a,z)=FALSE by MARGREL1:39;
then A7:'not' Pj(b,z)=TRUE or Pj(a,z)=TRUE by A2,A5,BINARITH:7;
A8:Pj((a '&' c) 'eqv' (b '&' c),z)
=Pj(((a '&' c) 'imp' (b '&' c)) '&' ((b '&' c) 'imp' (a '&' c)),z)
by BVFUNC_4:7
.=Pj((a '&' c) 'imp' (b '&' c),z) '&'
Pj((b '&' c) 'imp' (a '&' c),z)
by VALUAT_1:def 6
.=('not' Pj(a '&' c,z) 'or' Pj(b '&' c,z)) '&'
Pj((b '&' c) 'imp' (a '&' c),z)
by BVFUNC_1:def 11
.=('not' Pj(a '&' c,z) 'or' Pj(b '&' c,z)) '&'
('not' Pj(b '&' c,z) 'or' Pj(a '&' c,z))
by BVFUNC_1:def 11
.=('not'( Pj(a,z) '&' Pj(c,z)) 'or' Pj(b '&' c,z)) '&'
('not' Pj(b '&' c,z) 'or' Pj(a '&' c,z))
by VALUAT_1:def 6
.=('not'( Pj(a,z) '&' Pj(c,z)) 'or' (Pj(b,z) '&' Pj(c,z))) '&'
('not' Pj(b '&' c,z) 'or' Pj(a '&' c,z))
by VALUAT_1:def 6
.=('not'( Pj(a,z) '&' Pj(c,z)) 'or' (Pj(b,z) '&' Pj(c,z))) '&'
('not'( Pj(b,z) '&' Pj(c,z)) 'or' Pj(a '&' c,z))
by VALUAT_1:def 6
.=('not'( Pj(a,z) '&' Pj(c,z)) 'or' (Pj(b,z) '&' Pj(c,z))) '&'
('not'( Pj(b,z) '&' Pj(c,z)) 'or' (Pj(a,z) '&' Pj(c,z)))
by VALUAT_1:def 6
.=(('not' Pj(a,z) 'or' 'not' Pj(c,z)) 'or' (Pj(b,z) '&' Pj(c,z))) '&'
('not'( Pj(b,z) '&' Pj(c,z)) 'or' (Pj(a,z) '&' Pj(c,z)))
by BINARITH:9
.=(('not' Pj(a,z) 'or' 'not' Pj(c,z)) 'or' (Pj(b,z) '&' Pj(c,z))) '&'
(('not' Pj(b,z) 'or' 'not' Pj(c,z)) 'or' (Pj(a,z) '&' Pj(c,z)))
by BINARITH:9;
now per cases by A3,A4,BINARITH:7;
case A9:'not' Pj(a,z)=TRUE;
then A10:Pj(a,z)=FALSE by MARGREL1:41;
then 'not' Pj(b,z)=TRUE by A6,BINARITH:7;
then Pj(b,z)=FALSE by MARGREL1:41;
then Pj((a '&' c) 'eqv' (b '&' c),z)
=((TRUE 'or' 'not' Pj(c,z)) 'or' FALSE) '&'
((TRUE 'or' 'not' Pj(c,z)) 'or' (FALSE '&' Pj(c,z)))
by A8,A9,A10,MARGREL1:49
.=((TRUE 'or' 'not' Pj(c,z)) 'or' FALSE) '&'
((TRUE 'or' 'not' Pj(c,z)) 'or' FALSE)
by MARGREL1:49
.=(TRUE 'or' 'not' Pj(c,z)) '&' ((TRUE 'or' 'not'
Pj(c,z)) 'or' FALSE) by BINARITH:7
.=(TRUE 'or' 'not' Pj(c,z)) '&' (TRUE 'or' 'not' Pj(c,z)) by BINARITH:7
.=TRUE '&' (TRUE 'or' 'not' Pj(c,z)) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
case A11:Pj(b,z)=TRUE;
then 'not' Pj(b,z)=FALSE by MARGREL1:41;
then Pj((a '&' c) 'eqv' (b '&' c),z)
=((FALSE 'or' 'not' Pj(c,z)) 'or' Pj(c,z)) '&'
((FALSE 'or' 'not'
Pj(c,z)) 'or' (TRUE '&' Pj(c,z))) by A7,A8,A11,MARGREL1:43,50
.=((FALSE 'or' 'not' Pj(c,z)) 'or' Pj(c,z)) '&'
((FALSE 'or' 'not' Pj(c,z)) 'or' Pj(c,z)) by MARGREL1:50
.=('not' Pj(c,z) 'or' Pj(c,z)) '&'
((FALSE 'or' 'not' Pj(c,z)) 'or' Pj(c,z)) by BINARITH:7
.=('not' Pj(c,z) 'or' Pj(c,z)) '&' ('not' Pj(c,z) 'or' Pj(c,z))
by BINARITH:7
.=TRUE '&' ('not' Pj(c,z) 'or' Pj(c,z)) by BINARITH:18
.=TRUE '&' TRUE by BINARITH:18
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis;
end;
theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'eqv' b) '<' (a 'or' c) 'eqv' (b 'or' c)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a 'eqv' b,z)=TRUE;
Pj(a 'eqv' b,z)
=Pj((a 'imp' b) '&' (b 'imp' a),z) by BVFUNC_4:7
.=Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z) by VALUAT_1:def 6;
then A2:Pj(a 'imp' b,z)=TRUE & Pj(b 'imp' a,z)=TRUE by A1,MARGREL1:45;
then A3:'not' Pj(a,z) 'or' Pj(b,z) = TRUE by BVFUNC_1:def 11;
A4: Pj(b,z)=TRUE or Pj(b,z)=FALSE by MARGREL1:39;
A5: Pj(b 'imp' a,z) = 'not' Pj(b,z) 'or' Pj(a,z) by BVFUNC_1:def 11;
A6:'not' Pj(b,z) 'or' Pj(a,z) = TRUE by A2,BVFUNC_1:def 11;
Pj(a,z)=TRUE or Pj(a,z)=FALSE by MARGREL1:39;
then A7:'not' Pj(b,z)=TRUE or Pj(a,z)=TRUE by A2,A5,BINARITH:7;
A8:Pj((a 'or' c) 'eqv' (b 'or' c),z)
=Pj(((a 'or' c) 'imp' (b 'or' c)) '&' ((b 'or' c) 'imp' (a 'or' c)),z)
by BVFUNC_4:7
.=Pj((a 'or' c) 'imp' (b 'or' c),z) '&'
Pj((b 'or' c) 'imp' (a 'or' c),z)
by VALUAT_1:def 6
.=('not' Pj(a 'or' c,z) 'or' Pj(b 'or' c,z)) '&'
Pj((b 'or' c) 'imp' (a 'or' c),z)
by BVFUNC_1:def 11
.=('not' Pj(a 'or' c,z) 'or' Pj(b 'or' c,z)) '&'
('not' Pj(b 'or' c,z) 'or' Pj(a 'or' c,z))
by BVFUNC_1:def 11
.=('not'( Pj(a,z) 'or' Pj(c,z)) 'or' Pj(b 'or' c,z)) '&'
('not' Pj(b 'or' c,z) 'or' Pj(a 'or' c,z))
by BVFUNC_1:def 7
.=('not'( Pj(a,z) 'or' Pj(c,z)) 'or' (Pj(b,z) 'or' Pj(c,z))) '&'
('not' Pj(b 'or' c,z) 'or' Pj(a 'or' c,z))
by BVFUNC_1:def 7
.=('not'( Pj(a,z) 'or' Pj(c,z)) 'or' (Pj(b,z) 'or' Pj(c,z))) '&'
('not'( Pj(b,z) 'or' Pj(c,z)) 'or' Pj(a 'or' c,z))
by BVFUNC_1:def 7
.=('not'( Pj(a,z) 'or' Pj(c,z)) 'or' (Pj(b,z) 'or' Pj(c,z))) '&'
('not'( Pj(b,z) 'or' Pj(c,z)) 'or' (Pj(a,z) 'or' Pj(c,z)))
by BVFUNC_1:def 7
.=(('not' Pj(a,z) '&' 'not' Pj(c,z)) 'or' (Pj(b,z) 'or' Pj(c,z))) '&'
('not'( Pj(b,z) 'or' Pj(c,z)) 'or' (Pj(a,z) 'or' Pj(c,z)))
by BINARITH:10
.=(('not' Pj(a,z) '&' 'not' Pj(c,z)) 'or' (Pj(b,z) 'or' Pj(c,z))) '&'
(('not' Pj(b,z) '&' 'not' Pj(c,z)) 'or' (Pj(a,z) 'or' Pj(c,z)))
by BINARITH:10;
now per cases by A3,A4,BINARITH:7;
case A9:'not' Pj(a,z)=TRUE;
then A10:Pj(a,z)=FALSE by MARGREL1:41;
then 'not' Pj(b,z)=TRUE by A6,BINARITH:7;
then Pj(b,z)=FALSE by MARGREL1:41;
then Pj((a 'or' c) 'eqv' (b 'or' c),z)
=('not' Pj(c,z) 'or' (FALSE 'or' Pj(c,z))) '&'
((TRUE '&' 'not' Pj(c,z)) 'or' (FALSE 'or' Pj(c,z)))
by A8,A9,A10,MARGREL1:50
.=('not' Pj(c,z) 'or' (FALSE 'or' Pj(c,z))) '&'
('not' Pj(c,z) 'or' (FALSE 'or' Pj(c,z)))
by MARGREL1:50
.=('not' Pj(c,z) 'or' Pj(c,z)) '&'
('not' Pj(c,z) 'or' (FALSE 'or' Pj(c,z)))
by BINARITH:7
.=('not' Pj(c,z) 'or' Pj(c,z)) '&' ('not' Pj(c,z) 'or' Pj(c,z))
by BINARITH:7
.=TRUE '&' ('not' Pj(c,z) 'or' Pj(c,z))
by BINARITH:18
.=TRUE '&' TRUE
by BINARITH:18
.=TRUE by MARGREL1:45;
hence thesis;
case A11:Pj(b,z)=TRUE;
then 'not' Pj(b,z)=FALSE by MARGREL1:41;
then Pj((a 'or' c) 'eqv' (b 'or' c),z)
=(FALSE 'or' (TRUE 'or' Pj(c,z))) '&'
((FALSE '&' 'not' Pj(c,z)) 'or' (TRUE 'or' Pj(c,z)))
by A7,A8,A11,MARGREL1:43,49
.=(FALSE 'or' (TRUE 'or' Pj(c,z))) '&'
(FALSE 'or' (TRUE 'or' Pj(c,z)))
by MARGREL1:49
.=(TRUE 'or' Pj(c,z)) '&'
(FALSE 'or' (TRUE 'or' Pj(c,z)))
by BINARITH:7
.=(TRUE 'or' Pj(c,z)) '&' (TRUE 'or' Pj(c,z))
by BINARITH:7
.=TRUE '&' (TRUE 'or' Pj(c,z))
by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis;
end;
theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a '<' (a 'eqv' b) 'eqv' (b 'eqv' a) 'eqv' a
proof
let a,b be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a,z)=TRUE;
then A2:'not' Pj(a,z)=FALSE by MARGREL1:41;
A3:Pj('not' a,z)='not' Pj(a,z) by VALUAT_1:def 5;
A4:Pj((a 'eqv' b) 'eqv' (b 'eqv' a),z)
=Pj(((a 'eqv' b) 'imp' (b 'eqv' a)) '&'
((b 'eqv' a) 'imp' (a 'eqv' b)),z)
by BVFUNC_4:7
.=Pj((a 'eqv' b) 'imp' (b 'eqv' a),z) '&'
Pj((b 'eqv' a) 'imp' (a 'eqv' b),z)
by VALUAT_1:def 6
.=Pj('not'( a 'eqv' b) 'or' (b 'eqv' a),z) '&'
Pj((b 'eqv' a) 'imp' (a 'eqv' b),z)
by BVFUNC_4:8
.=Pj('not'( a 'eqv' b) 'or' (b 'eqv' a),z) '&'
Pj('not'( b 'eqv' a) 'or' (a 'eqv' b),z)
by BVFUNC_4:8
.=(Pj('not'( a 'eqv' b),z) 'or' Pj(b 'eqv' a,z)) '&'
Pj('not'( b 'eqv' a) 'or' (a 'eqv' b),z)
by BVFUNC_1:def 7
.=(Pj('not'( a 'eqv' b),z) 'or' Pj(b 'eqv' a,z)) '&'
(Pj('not'( b 'eqv' a),z) 'or' Pj(a 'eqv' b,z))
by BVFUNC_1:def 7
.=(Pj('not'( (a 'imp' b) '&' (b 'imp' a)),z) 'or' Pj(b 'eqv' a,z)) '&'
(Pj('not'( b 'eqv' a),z) 'or' Pj(a 'eqv' b,z))
by BVFUNC_4:7
.=(Pj('not'( (a 'imp' b) '&' (b 'imp' a)),z) 'or' Pj(b 'eqv' a,z)) '&'
(Pj('not'( b 'eqv' a),z) 'or' Pj((a 'imp' b) '&' (b 'imp' a),z))
by BVFUNC_4:7
.=(Pj('not'( (a 'imp' b) '&' (b 'imp' a)),z) 'or'
Pj((b 'imp' a) '&' (a 'imp' b),z)) '&'
(Pj('not'( b 'eqv' a),z) 'or' Pj((a 'imp' b) '&' (b 'imp' a),z))
by BVFUNC_4:7
.=(Pj('not'( (a 'imp' b) '&' (b 'imp' a)),z) 'or'
Pj((b 'imp' a) '&' (a 'imp' b),z)) '&'
(Pj('not'( (b 'imp' a) '&' (a 'imp' b)),z) 'or'
Pj((a 'imp' b) '&' (b 'imp' a),z))
by BVFUNC_4:7
.=('not' Pj((a 'imp' b) '&' (b 'imp' a),z) 'or'
Pj((b 'imp' a) '&' (a 'imp' b),z)) '&'
(Pj('not'( (b 'imp' a) '&' (a 'imp' b)),z) 'or'
Pj((a 'imp' b) '&' (b 'imp' a),z))
by VALUAT_1:def 5
.=('not' Pj((a 'imp' b) '&' (b 'imp' a),z) 'or'
Pj((b 'imp' a) '&' (a 'imp' b),z)) '&'
('not' Pj((b 'imp' a) '&' (a 'imp' b),z) 'or'
Pj((a 'imp' b) '&' (b 'imp' a),z))
by VALUAT_1:def 5
.=('not'( Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)) 'or'
Pj((b 'imp' a) '&' (a 'imp' b),z)) '&'
('not' Pj((b 'imp' a) '&' (a 'imp' b),z) 'or'
Pj((a 'imp' b) '&' (b 'imp' a),z))
by VALUAT_1:def 6
.=('not'( Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)) 'or'
(Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z))) '&'
('not' Pj((b 'imp' a) '&' (a 'imp' b),z) 'or'
Pj((a 'imp' b) '&' (b 'imp' a),z))
by VALUAT_1:def 6
.=('not'( Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)) 'or'
(Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z))) '&'
('not'( Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z)) 'or'
Pj((a 'imp' b) '&' (b 'imp' a),z))
by VALUAT_1:def 6
.=('not'( Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)) 'or'
(Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z))) '&'
('not'( Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z)) 'or'
(Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)))
by VALUAT_1:def 6
.=('not'( Pj('not' a 'or' b,z) '&' Pj(b 'imp' a,z)) 'or'
(Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z))) '&'
('not'( Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z)) 'or'
(Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)))
by BVFUNC_4:8
.=('not'( Pj('not' a 'or' b,z) '&' Pj('not' b 'or' a,z)) 'or'
(Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z))) '&'
('not'( Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z)) 'or'
(Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)))
by BVFUNC_4:8
.=('not'( Pj('not' a 'or' b,z) '&' Pj('not' b 'or' a,z)) 'or'
(Pj('not' b 'or' a,z) '&' Pj(a 'imp' b,z))) '&'
('not'( Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z)) 'or'
(Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)))
by BVFUNC_4:8
.=('not'( Pj('not' a 'or' b,z) '&' Pj('not' b 'or' a,z)) 'or'
(Pj('not' b 'or' a,z) '&' Pj('not' a 'or' b,z))) '&'
('not'( Pj(b 'imp' a,z) '&' Pj(a 'imp' b,z)) 'or'
(Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)))
by BVFUNC_4:8
.=('not'( Pj('not' a 'or' b,z) '&' Pj('not' b 'or' a,z)) 'or'
(Pj('not' b 'or' a,z) '&' Pj('not' a 'or' b,z))) '&'
('not'( Pj('not' b 'or' a,z) '&' Pj(a 'imp' b,z)) 'or'
(Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)))
by BVFUNC_4:8
.=('not'( Pj('not' a 'or' b,z) '&' Pj('not' b 'or' a,z)) 'or'
(Pj('not' b 'or' a,z) '&' Pj('not' a 'or' b,z))) '&'
('not'( Pj('not' b 'or' a,z) '&' Pj('not' a 'or' b,z)) 'or'
(Pj(a 'imp' b,z) '&' Pj(b 'imp' a,z)))
by BVFUNC_4:8
.=('not'( Pj('not' a 'or' b,z) '&' Pj('not' b 'or' a,z)) 'or'
(Pj('not' b 'or' a,z) '&' Pj('not' a 'or' b,z))) '&'
('not'( Pj('not' b 'or' a,z) '&' Pj('not' a 'or' b,z)) 'or'
(Pj('not' a 'or' b,z) '&' Pj(b 'imp' a,z)))
by BVFUNC_4:8
.=('not'( Pj('not' a 'or' b,z) '&' Pj('not' b 'or' a,z)) 'or'
(Pj('not' b 'or' a,z) '&' Pj('not' a 'or' b,z))) '&'
('not'( Pj('not' b 'or' a,z) '&' Pj('not' a 'or' b,z)) 'or'
(Pj('not' a 'or' b,z) '&' Pj('not' b 'or' a,z)))
by BVFUNC_4:8
.=('not'( (Pj('not' a,z) 'or' Pj(b,z)) '&' (Pj('not' b 'or' a,z))) 'or'
((Pj('not' b 'or' a,z)) '&' (Pj('not' a 'or' b,z)))) '&'
('not'( (Pj('not' b 'or' a,z)) '&' (Pj('not' a 'or' b,z))) 'or'
((Pj('not' a 'or' b,z)) '&' (Pj('not' b 'or' a,z))))
by BVFUNC_1:def 7
.=('not'( (Pj('not' a,z) 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' Pj(a,z))) 'or'
((Pj('not' b 'or' a,z)) '&' (Pj('not' a 'or' b,z)))) '&'
('not'( (Pj('not' b 'or' a,z)) '&' (Pj('not' a 'or' b,z))) 'or'
((Pj('not' a 'or' b,z)) '&' (Pj('not' b 'or' a,z))))
by BVFUNC_1:def 7
.=('not'( (Pj('not' a,z) 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' Pj(a,z))) 'or'
((Pj('not' b,z) 'or' Pj(a,z)) '&' (Pj('not' a 'or' b,z)))) '&'
('not'( (Pj('not' b 'or' a,z)) '&' (Pj('not' a 'or' b,z))) 'or'
((Pj('not' a 'or' b,z)) '&' (Pj('not' b 'or' a,z))))
by BVFUNC_1:def 7
.=('not'( (Pj('not' a,z) 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' Pj(a,z))) 'or'
((Pj('not' b,z) 'or' Pj(a,z)) '&' (Pj('not' a,z) 'or' Pj(b,z)))) '&'
('not'( (Pj('not' b 'or' a,z)) '&' (Pj('not' a 'or' b,z))) 'or'
((Pj('not' a 'or' b,z)) '&' (Pj('not' b 'or' a,z))))
by BVFUNC_1:def 7
.=('not'( (Pj('not' a,z) 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' Pj(a,z))) 'or'
((Pj('not' b,z) 'or' Pj(a,z)) '&' (Pj('not' a,z) 'or' Pj(b,z)))) '&'
('not'( (Pj('not' b,z) 'or' Pj(a,z)) '&' (Pj('not' a 'or' b,z))) 'or'
((Pj('not' a 'or' b,z)) '&' (Pj('not' b 'or' a,z))))
by BVFUNC_1:def 7
.=('not'( (Pj('not' a,z) 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' Pj(a,z))) 'or'
((Pj('not' b,z) 'or' Pj(a,z)) '&' (Pj('not' a,z) 'or' Pj(b,z)))) '&'
('not'( (Pj('not' b,z) 'or' Pj(a,z)) '&' (Pj('not' a,z) 'or' Pj(b,z))) 'or'
((Pj('not' a 'or' b,z)) '&' (Pj('not' b 'or' a,z))))
by BVFUNC_1:def 7
.=('not'( (Pj('not' a,z) 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' Pj(a,z))) 'or'
((Pj('not' b,z) 'or' Pj(a,z)) '&' (Pj('not' a,z) 'or' Pj(b,z)))) '&'
('not'( (Pj('not' b,z) 'or' Pj(a,z)) '&' (Pj('not' a,z) 'or' Pj(b,z))) 'or'
((Pj('not' a,z) 'or' Pj(b,z)) '&' (Pj('not' b 'or' a,z))))
by BVFUNC_1:def 7
.=('not'( (FALSE 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' TRUE)) 'or'
((Pj('not' b,z) 'or' TRUE) '&' (FALSE 'or' Pj(b,z)))) '&'
('not'( (Pj('not' b,z) 'or' TRUE) '&' (FALSE 'or' Pj(b,z))) 'or'
((FALSE 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' TRUE)))
by A1,A2,A3,BVFUNC_1:def 7
.=('not'( Pj(b,z) '&' (Pj('not' b,z) 'or' TRUE)) 'or'
((Pj('not' b,z) 'or' TRUE) '&' (FALSE 'or' Pj(b,z)))) '&'
('not'( (Pj('not' b,z) 'or' TRUE) '&' (FALSE 'or' Pj(b,z))) 'or'
((FALSE 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' TRUE)))
by BINARITH:7
.=('not'( Pj(b,z) '&' (Pj('not' b,z) 'or' TRUE)) 'or'
((Pj('not' b,z) 'or' TRUE) '&' Pj(b,z))) '&'
('not'( (Pj('not' b,z) 'or' TRUE) '&' (FALSE 'or' Pj(b,z))) 'or'
((FALSE 'or' Pj(b,z)) '&' (Pj('not' b,z) 'or' TRUE)))
by BINARITH:7
.=('not'( Pj(b,z) '&' (Pj('not' b,z) 'or' TRUE)) 'or'
((Pj('not' b,z) 'or' TRUE) '&' Pj(b,z))) '&'
('not'( (Pj('not' b,z) 'or' TRUE) '&' (FALSE 'or' Pj(b,z))) 'or'
(Pj(b,z) '&' (Pj('not' b,z) 'or' TRUE)))
by BINARITH:7
.=('not'( Pj(b,z) '&' TRUE) 'or'
((Pj('not' b,z) 'or' TRUE) '&' Pj(b,z))) '&'
('not'( (Pj('not' b,z) 'or' TRUE) '&' (FALSE 'or' Pj(b,z))) 'or'
(Pj(b,z) '&' (Pj('not' b,z) 'or' TRUE)))
by BINARITH:19
.=('not'( Pj(b,z) '&' TRUE) 'or' (TRUE '&' Pj(b,z))) '&'
('not'( (Pj('not' b,z) 'or' TRUE) '&' (FALSE 'or' Pj(b,z))) 'or'
(Pj(b,z) '&' (Pj('not' b,z) 'or' TRUE)))
by BINARITH:19
.=('not'( Pj(b,z) '&' TRUE) 'or' (TRUE '&' Pj(b,z))) '&'
('not'( TRUE '&' (FALSE 'or' Pj(b,z))) 'or'
(Pj(b,z) '&' (Pj('not' b,z) 'or' TRUE)))
by BINARITH:19
.=('not'( Pj(b,z) '&' TRUE) 'or' (TRUE '&' Pj(b,z))) '&'
('not'( TRUE '&' (FALSE 'or' Pj(b,z))) 'or'
(Pj(b,z) '&' TRUE))
by BINARITH:19
.=('not'( TRUE '&' Pj(b,z)) 'or' (TRUE '&' Pj(b,z))) '&'
('not'( TRUE '&' Pj(b,z)) 'or' (Pj(b,z) '&' TRUE)) by BINARITH:7
.=('not' Pj(b,z) 'or' (TRUE '&' Pj(b,z))) '&'
('not'( TRUE '&' Pj(b,z)) 'or' (TRUE '&' Pj(b,z)))
by MARGREL1:50
.=('not' Pj(b,z) 'or' (TRUE '&' Pj(b,z))) '&'
('not' Pj(b,z) 'or' (TRUE '&' Pj(b,z)))
by MARGREL1:50
.=('not' Pj(b,z) 'or' Pj(b,z)) '&'
('not' Pj(b,z) 'or' (TRUE '&' Pj(b,z)))
by MARGREL1:50
.=('not' Pj(b,z) 'or' Pj(b,z)) '&' ('not' Pj(b,z) 'or' Pj(b,z))
by MARGREL1:50
.=TRUE '&' ('not' Pj(b,z) 'or' Pj(b,z))
by BINARITH:18
.=TRUE '&' TRUE
by BINARITH:18
.=TRUE by MARGREL1:45;
Pj((a 'eqv' b) 'eqv' (b 'eqv' a) 'eqv' a,z)
=Pj((((a 'eqv' b) 'eqv' (b 'eqv' a)) 'imp' a) '&'
(a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a))),z)
by BVFUNC_4:7
.=Pj(((a 'eqv' b) 'eqv' (b 'eqv' a)) 'imp' a,z) '&'
Pj(a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a)),z)
by VALUAT_1:def 6
.=('not' Pj((a 'eqv' b) 'eqv' (b 'eqv' a),z) 'or' Pj(a,z)) '&'
Pj(a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a)),z)
by BVFUNC_1:def 11
.=('not' Pj((a 'eqv' b) 'eqv' (b 'eqv' a),z) 'or' Pj(a,z)) '&'
('not' Pj(a,z) 'or' Pj((a 'eqv' b) 'eqv' (b 'eqv' a),z))
by BVFUNC_1:def 11
.=(FALSE 'or' Pj(a,z)) '&' ('not' Pj(a,z) 'or' TRUE)
by A4,MARGREL1:41
.=Pj(a,z) '&' ('not' Pj(a,z) 'or' TRUE)
by BINARITH:7
.=TRUE '&' Pj(a,z) by BINARITH:19
.=TRUE by A1,MARGREL1:50;
hence thesis;
end;
theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a '<' (a 'imp' b) 'eqv' b
proof
let a,b be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a,z)=TRUE;
then A2:'not' Pj(a,z)=FALSE by MARGREL1:41;
Pj((a 'imp' b) 'eqv' b,z)
=Pj(('not' a 'or' b) 'eqv' b,z) by BVFUNC_4:8
.=Pj((('not' a 'or' b) 'imp' b) '&' (b 'imp' ('not' a 'or' b)),z)
by BVFUNC_4:7
.=Pj(('not'( 'not' a 'or' b) 'or' b) '&' (b 'imp' ('not'
a 'or' b)),z) by BVFUNC_4:8
.=Pj(('not'( 'not' a 'or' b) 'or' b) '&' ('not' b 'or' ('not'
a 'or' b)),z) by BVFUNC_4:8
.=Pj('not'( 'not' a 'or' b) 'or' b,z) '&' Pj('not' b 'or' ('not' a 'or' b),z)
by VALUAT_1:def 6
.=(Pj('not'( 'not' a 'or' b),z) 'or' Pj(b,z)) '&' Pj('not' b 'or' ('not'
a 'or' b),z)
by BVFUNC_1:def 7
.=('not' Pj('not' a 'or' b,z) 'or' Pj(b,z)) '&' Pj('not' b 'or' ('not'
a 'or' b),z)
by VALUAT_1:def 5
.=('not'( Pj('not' a,z) 'or' Pj(b,z)) 'or' Pj(b,z)) '&'
Pj('not' b 'or' ('not' a 'or' b),z)
by BVFUNC_1:def 7
.=('not'( 'not' Pj(a,z) 'or' Pj(b,z)) 'or' Pj(b,z)) '&'
Pj('not' b 'or' ('not' a 'or' b),z)
by VALUAT_1:def 5
.=(('not' 'not' Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj(b,z)) '&'
Pj('not' b 'or' ('not' a 'or' b),z)
by BINARITH:10
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj(b,z)) '&' Pj('not' b 'or' ('not'
a 'or' b),z)
by MARGREL1:40
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj(b,z)) '&'
(Pj('not' b,z) 'or' Pj('not' a 'or' b,z))
by BVFUNC_1:def 7
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj(b,z)) '&'
(Pj('not' b,z) 'or' (Pj('not' a,z) 'or' Pj(b,z)))
by BVFUNC_1:def 7
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj(b,z)) '&'
(Pj('not' b,z) 'or' ('not' Pj(a,z) 'or' Pj(b,z)))
by VALUAT_1:def 5
.=((TRUE '&' 'not' Pj(b,z)) 'or' Pj(b,z)) '&'
('not' Pj(b,z) 'or' (FALSE 'or' Pj(b,z)))
by A1,A2,VALUAT_1:def 5
.=('not' Pj(b,z) 'or' Pj(b,z)) '&'
('not' Pj(b,z) 'or' (FALSE 'or' Pj(b,z)))
by MARGREL1:50
.=('not' Pj(b,z) 'or' Pj(b,z)) '&'
('not' Pj(b,z) 'or' Pj(b,z))
by BINARITH:7
.=TRUE '&' ('not' Pj(b,z) 'or' Pj(b,z))
by BINARITH:18
.=TRUE '&' TRUE
by BINARITH:18
.=TRUE by MARGREL1:45;
hence thesis;
end;
theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a '<' (b 'imp' a) 'eqv' a
proof
let a,b be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a,z)=TRUE;
then A2:'not' Pj(a,z)=FALSE by MARGREL1:41;
Pj((b 'imp' a) 'eqv' a,z)
=Pj(('not' b 'or' a) 'eqv' a,z) by BVFUNC_4:8
.=Pj((('not' b 'or' a) 'imp' a) '&' (a 'imp' ('not' b 'or' a)),z)
by BVFUNC_4:7
.=Pj(('not'( 'not' b 'or' a) 'or' a) '&' (a 'imp' ('not'
b 'or' a)),z) by BVFUNC_4:8
.=Pj(('not'( 'not' b 'or' a) 'or' a) '&' ('not' a 'or' ('not'
b 'or' a)),z) by BVFUNC_4:8
.=Pj('not'( 'not' b 'or' a) 'or' a,z) '&' Pj('not' a 'or' ('not' b 'or' a),z)
by VALUAT_1:def 6
.=(Pj('not'( 'not' b 'or' a),z) 'or' Pj(a,z)) '&' Pj('not' a 'or' ('not'
b 'or' a),z)
by BVFUNC_1:def 7
.=('not' Pj('not' b 'or' a,z) 'or' Pj(a,z)) '&' Pj('not' a 'or' ('not'
b 'or' a),z) by VALUAT_1:def 5
.=('not'( Pj('not' b,z) 'or' Pj(a,z)) 'or' Pj(a,z)) '&'
Pj('not' a 'or' ('not' b 'or' a),z)
by BVFUNC_1:def 7
.=('not'( 'not' Pj(b,z) 'or' Pj(a,z)) 'or' Pj(a,z)) '&'
Pj('not' a 'or' ('not' b 'or' a),z)
by VALUAT_1:def 5
.=(('not' 'not' Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj(a,z)) '&'
Pj('not' a 'or' ('not' b 'or' a),z)
by BINARITH:10
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj(a,z)) '&' Pj('not' a 'or' ('not'
b 'or' a),z)
by MARGREL1:40
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj(a,z)) '&'
(Pj('not' a,z) 'or' Pj('not' b 'or' a,z))
by BVFUNC_1:def 7
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj(a,z)) '&'
(Pj('not' a,z) 'or' (Pj('not' b,z) 'or' Pj(a,z)))
by BVFUNC_1:def 7
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj(a,z)) '&'
(Pj('not' a,z) 'or' ('not' Pj(b,z) 'or' Pj(a,z)))
by VALUAT_1:def 5
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj(a,z)) '&'
('not' Pj(a,z) 'or' ('not' Pj(b,z) 'or' Pj(a,z)))
by VALUAT_1:def 5
.=TRUE '&' (FALSE 'or' ('not' Pj(b,z) 'or' TRUE))
by A1,A2,BINARITH:19
.=FALSE 'or' ('not' Pj(b,z) 'or' TRUE)
by MARGREL1:50
.='not' Pj(b,z) 'or' TRUE
by BINARITH:7
.=TRUE by BINARITH:19;
hence thesis;
end;
theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a '<' (a '&' b) 'eqv' (b '&' a) 'eqv' a
proof
let a,b be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a,z)=TRUE;
A2:Pj((a '&' b) 'eqv' (a '&' b),z)
=Pj(((a '&' b) 'imp' (a '&' b)) '&' ((a '&' b) 'imp' (a '&' b)),z)
by BVFUNC_4:7
.=Pj((a '&' b) 'imp' (a '&' b),z) '&' Pj((a '&' b) 'imp' (a '&' b),z)
by VALUAT_1:def 6
.=Pj((a '&' b) 'imp' (a '&' b),z)
by BINARITH:16
.=Pj('not'( a '&' b) 'or' (a '&' b),z)
by BVFUNC_4:8
.=Pj(I_el(Y),z) by BVFUNC_4:6
.=TRUE by BVFUNC_1:def 14;
Pj((a '&' b) 'eqv' (b '&' a) 'eqv' a,z)
=Pj((((a '&' b) 'eqv' (a '&' b)) 'imp' a) '&'
(a 'imp' ((a '&' b) 'eqv' (a '&' b))),z) by BVFUNC_4:7
.=Pj(((a '&' b) 'eqv' (a '&' b)) 'imp' a,z) '&'
Pj(a 'imp' ((a '&' b) 'eqv' (a '&' b)),z)
by VALUAT_1:def 6
.=Pj('not'( (a '&' b) 'eqv' (a '&' b)) 'or' a,z) '&'
Pj(a 'imp' ((a '&' b) 'eqv' (a '&' b)),z)
by BVFUNC_4:8
.=Pj('not'( (a '&' b) 'eqv' (a '&' b)) 'or' a,z) '&'
Pj('not' a 'or' ((a '&' b) 'eqv' (a '&' b)),z)
by BVFUNC_4:8
.=(Pj('not'( (a '&' b) 'eqv' (a '&' b)),z) 'or' Pj(a,z)) '&'
Pj('not' a 'or' ((a '&' b) 'eqv' (a '&' b)),z)
by BVFUNC_1:def 7
.=(Pj('not'( (a '&' b) 'eqv' (a '&' b)),z) 'or' Pj(a,z)) '&'
(Pj('not' a,z) 'or' Pj((a '&' b) 'eqv' (a '&' b),z))
by BVFUNC_1:def 7
.=('not' Pj((a '&' b) 'eqv' (a '&' b),z) 'or' Pj(a,z)) '&'
(Pj('not' a,z) 'or' Pj((a '&' b) 'eqv' (a '&' b),z))
by VALUAT_1:def 5
.=(FALSE 'or' Pj(a,z)) '&' (Pj('not' a,z) 'or' TRUE)
by A2,MARGREL1:41
.=Pj(a,z) '&' (Pj('not' a,z) 'or' TRUE)
by BINARITH:7
.=TRUE '&' Pj(a,z) by BINARITH:19
.=TRUE by A1,MARGREL1:50;
hence thesis;
end;