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
consider k3 being Function such that
A1: (b 'imp' c) 'imp' ((a 'imp' b) '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 ((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

A5: 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;

A6: 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;

A7: 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;

((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 8
.= ('not' ((b 'imp' c) . x)) 'or' (('not' ((a 'imp' b) . x)) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 8
.= ('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' ((a 'imp' b) . x)) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 8
.= ('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' (('not' (a . x)) 'or' (b . x))) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 8
.= ((b . x) '&' ('not' (c . x))) 'or' ((('not' ('not' (a . x))) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) by BVFUNC_1:def 8
.= ((((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:11
.= (((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 A7, BINARITH:11
.= (((a . x) '&' ('not' (b . x))) 'or' TRUE) '&' ((((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x)) by BINARITH:10
.= TRUE '&' ((((a . x) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x)) by BINARITH:10
.= ((('not' (b . x)) '&' (a . x)) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (b . x) by MARGREL1:14
.= (((('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:11
.= (((('not' (a . x)) 'or' (c . x)) 'or' ('not' (b . x))) '&' TRUE) 'or' (b . x) by A5, BINARITH:10
.= ((('not' (a . x)) 'or' (c . x)) 'or' ('not' (b . x))) 'or' (b . x) by MARGREL1:14
.= (('not' (a . x)) 'or' (c . x)) 'or' TRUE by A6, BINARITH:11
.= TRUE by BINARITH:10 ;
hence ((b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c))) . x = (I_el Y) . x by 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 (b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c)) = I_el Y by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum