let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs (Y,BOOLEAN) holds (a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c)) = I_el Y
let a, b, c be Element of Funcs (Y,BOOLEAN); :: thesis: (a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c)) = I_el Y
consider k3 being Function such that
A1: (a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c)) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: I_el Y = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds ((a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c))) . x = (I_el Y) . x

proof

let x be Element of Y; :: thesis: ((a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c))) . x = (I_el Y) . x
A5: ((a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c))) . x = ('not' ((a 'imp' b) . x)) 'or' (((b 'imp' c) 'imp' (a 'imp' c)) . x) by BVFUNC_1:def 8
.= ('not' ((a 'imp' b) . x)) 'or' (('not' ((b 'imp' c) . x)) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 8 ;
A6: 'not' ((b 'imp' c) . x) = 'not' (('not' (b . x)) 'or' (c . x)) by BVFUNC_1:def 8
.= (b . x) '&' ('not' (c . x)) ;

A7: 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:10; :: 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 MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ('not' (b . x)) 'or' (b . x) = TRUE ; :: thesis: verum

end;

end;

end;

A8: (a 'imp' c) . x = ('not' (a . x)) 'or' (c . x) by BVFUNC_1:def 8;

A9: 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:10; :: 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 MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ('not' (c . x)) 'or' (c . x) = TRUE ; :: thesis: verum

end;

end;

end;

A10: 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:10; :: 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 MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ('not' (a . x)) 'or' (a . x) = TRUE ; :: thesis: verum

end;

end;

end;

'not' ((a 'imp' b) . x) = 'not' (('not' (a . x)) 'or' (b . x)) by BVFUNC_1:def 8
.= (a . x) '&' ('not' (b . x)) ;
then ('not' ((a 'imp' b) . x)) 'or' (('not' ((b 'imp' c) . x)) 'or' ((a 'imp' c) . x)) = ((((b . x) '&' ('not' (c . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (a . x)) '&' ((((b . x) '&' ('not' (c . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' ('not' (b . x))) by A6, A8, XBOOLEAN:9
.= (((b . x) '&' ('not' (c . x))) 'or' (((c . x) 'or' ('not' (a . x))) 'or' (a . x))) '&' ((((b . x) '&' ('not' (c . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' ('not' (b . x))) by BINARITH:11
.= (((b . x) '&' ('not' (c . x))) 'or' ((c . x) 'or' (('not' (a . x)) 'or' (a . x)))) '&' (((('not' (a . x)) 'or' (c . x)) 'or' ((b . x) '&' ('not' (c . x)))) 'or' ('not' (b . x))) by BINARITH:11
.= (((b . x) '&' ('not' (c . x))) 'or' ((c . x) 'or' (('not' (a . x)) 'or' (a . x)))) '&' ((('not' (a . x)) 'or' (c . x)) 'or' ((('not' (c . x)) '&' (b . x)) 'or' ('not' (b . x)))) by BINARITH:11
.= (((b . x) '&' ('not' (c . x))) 'or' ((c . x) 'or' (('not' (a . x)) 'or' (a . x)))) '&' ((('not' (a . x)) 'or' (c . x)) 'or' ((('not' (b . x)) 'or' ('not' (c . x))) '&' (('not' (b . x)) 'or' (b . x)))) by XBOOLEAN:9
.= (((b . x) '&' ('not' (c . x))) 'or' TRUE) '&' ((('not' (a . x)) 'or' (c . x)) 'or' ((('not' (b . x)) 'or' ('not' (c . x))) '&' TRUE)) by A7, A10, BINARITH:10
.= TRUE '&' ((('not' (a . x)) 'or' (c . x)) 'or' ((('not' (b . x)) 'or' ('not' (c . x))) '&' TRUE)) by BINARITH:10
.= (('not' (a . x)) 'or' (c . x)) 'or' (TRUE '&' (('not' (b . x)) 'or' ('not' (c . x)))) by MARGREL1:14
.= (('not' (a . x)) 'or' (c . x)) 'or' (('not' (c . x)) 'or' ('not' (b . x))) by MARGREL1:14
.= ((('not' (a . x)) 'or' (c . x)) 'or' ('not' (c . x))) 'or' ('not' (b . x)) by BINARITH:11
.= (('not' (a . x)) 'or' TRUE) 'or' ('not' (b . x)) by A9, BINARITH:11
.= TRUE 'or' ('not' (b . x)) by BINARITH:10
.= TRUE by BINARITH:10 ;
hence ((a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c))) . x = (I_el Y) . x by A5, BVFUNC_1:def 11; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence (a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c)) = I_el Y by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum