let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds (('not' a) 'imp' b) 'imp' (('not' b) 'imp' a) = I_el Y
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: (('not' a) 'imp' b) 'imp' (('not' b) 'imp' a) = I_el Y
A1: for x being Element of Y holds ((('not' a) 'imp' b) 'imp' (('not' b) 'imp' a)) . x = (I_el Y) . x

proof

let x be Element of Y; :: thesis: ((('not' a) 'imp' b) 'imp' (('not' b) 'imp' a)) . x = (I_el Y) . x

A2: now

per cases ( a . x = TRUE or a . x = FALSE ) by XBOOLEAN:def 3;

case a . x = TRUE ; :: thesis: ('not' (a . x)) 'or' (a . x) = TRUE

hence ('not' (a . x)) 'or' (a . x) = TRUE by BINARITH:19; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: ('not' (a . x)) 'or' (a . x) = TRUE

then ('not' (a . x)) 'or' (a . x) = TRUE 'or' FALSE by MARGREL1:41
.= TRUE by BINARITH:19 ;
hence ('not' (a . x)) 'or' (a . x) = TRUE ; :: thesis: verum

end;

end;

end;

A3: now

per cases ( b . x = TRUE or b . x = FALSE ) by XBOOLEAN:def 3;

case b . x = TRUE ; :: thesis: ('not' (b . x)) 'or' (b . x) = TRUE

hence ('not' (b . x)) 'or' (b . x) = TRUE by BINARITH:19; :: thesis: verum

end;

case b . x = FALSE ; :: thesis: ('not' (b . x)) 'or' (b . x) = TRUE

then ('not' (b . x)) 'or' (b . x) = TRUE 'or' FALSE by MARGREL1:41
.= TRUE by BINARITH:19 ;
hence ('not' (b . x)) 'or' (b . x) = TRUE ; :: thesis: verum

end;

end;

end;

((('not' a) 'imp' b) 'imp' (('not' b) 'imp' a)) . x = ('not' ((('not' a) 'imp' b) . x)) 'or' ((('not' b) 'imp' a) . x) by BVFUNC_1:def 11
.= ('not' (('not' (('not' a) . x)) 'or' (b . x))) 'or' ((('not' b) 'imp' a) . x) by BVFUNC_1:def 11
.= (('not' ('not' (('not' a) . x))) '&' ('not' (b . x))) 'or' (('not' (('not' b) . x)) 'or' (a . x)) by BVFUNC_1:def 11
.= (('not' (a . x)) '&' ('not' (b . x))) 'or' (('not' (('not' b) . x)) 'or' (a . x)) by MARGREL1:def 20
.= ((b . x) 'or' (a . x)) 'or' (('not' (a . x)) '&' ('not' (b . x))) by MARGREL1:def 20
.= (((b . x) 'or' (a . x)) 'or' ('not' (a . x))) '&' (((b . x) 'or' (a . x)) 'or' ('not' (b . x))) by XBOOLEAN:9
.= ((b . x) 'or' TRUE ) '&' (((b . x) 'or' (a . x)) 'or' ('not' (b . x))) by A2, BINARITH:20
.= TRUE '&' (((b . x) 'or' (a . x)) 'or' ('not' (b . x))) by BINARITH:19
.= ((a . x) 'or' (b . x)) 'or' ('not' (b . x)) by MARGREL1:50
.= (a . x) 'or' TRUE by A3, BINARITH:20
.= TRUE by BINARITH:19 ;
hence ((('not' a) 'imp' b) 'imp' (('not' b) 'imp' a)) . x = (I_el Y) . x by BVFUNC_1:def 14; :: thesis: verum

end;

consider k3 being Function such that
A4: ( (('not' a) 'imp' b) 'imp' (('not' b) 'imp' a) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A5: ( I_el Y = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
for u being set st u in Y holds
k3 . u = k4 . u by A1, A4, A5;
hence (('not' a) 'imp' b) 'imp' (('not' b) 'imp' a) = I_el Y by A4, A5, FUNCT_1:9; :: thesis: verum