let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds (b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c)) = I_el Y
let a, b, c be Element of Funcs Y,BOOLEAN ; :: thesis: (b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c)) = I_el Y
A1: for x being Element of Y holds ((b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c))) . x = (I_el Y) . x

proof

let x be Element of Y; :: thesis: ((b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c))) . 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;

A4: now

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

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

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

end;

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

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

end;

end;

end;

((b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c))) . x = ('not' ((b 'imp' c) . x)) 'or' (((a 'imp' b) 'imp' (a 'imp' c)) . x) by BVFUNC_1:def 11
.= ('not' ((b 'imp' c) . x)) 'or' (('not' ((a 'imp' b) . x)) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 11
.= ('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' ((a 'imp' b) . x)) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 11
.= ('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' (('not' (a . x)) 'or' (b . x))) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 11
.= ((b . x) '&' ('not' (c . x))) 'or' ((('not' ('not' (a . x))) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) by BVFUNC_1:def 11
.= ((((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' ('not' (c . x))) '&' ((((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x)) by XBOOLEAN:9
.= (((a . x) '&' ('not' (b . x))) 'or' ((('not' (a . x)) 'or' (c . x)) 'or' ('not' (c . x)))) '&' ((((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x)) by BINARITH:20
.= (((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' TRUE )) '&' ((((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x)) by A4, BINARITH:20
.= (((a . x) '&' ('not' (b . x))) 'or' TRUE ) '&' ((((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x)) by BINARITH:19
.= TRUE '&' ((((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x)) by BINARITH:19
.= ((('not' (b . x)) '&' (a . x)) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x) by MARGREL1:50
.= (((('not' (a . x)) 'or' (c . x)) 'or' ('not' (b . x))) '&' ((('not' (a . x)) 'or' (c . x)) 'or' (a . x))) 'or' (b . x) by XBOOLEAN:9
.= (((('not' (a . x)) 'or' (c . x)) 'or' ('not' (b . x))) '&' ((c . x) 'or' (('not' (a . x)) 'or' (a . x)))) 'or' (b . x) by BINARITH:20
.= (((('not' (a . x)) 'or' (c . x)) 'or' ('not' (b . x))) '&' TRUE ) 'or' (b . x) by A2, BINARITH:19
.= ((('not' (a . x)) 'or' (c . x)) 'or' ('not' (b . x))) 'or' (b . x) by MARGREL1:50
.= (('not' (a . x)) 'or' (c . x)) 'or' TRUE by A3, BINARITH:20
.= TRUE by BINARITH:19 ;
hence ((b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c))) . x = (I_el Y) . x by BVFUNC_1:def 14; :: thesis: verum

end;

consider k3 being Function such that
A5: ( (b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c)) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A6: ( 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, A5, A6;
hence (b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c)) = I_el Y by A5, A6, FUNCT_1:9; :: thesis: verum