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

proof

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

A2: now :: thesis: ( ( a . x = TRUE & ('not' (a . x)) 'or' (a . x) = TRUE ) or ( a . x = FALSE & ('not' (a . x)) 'or' (a . x) = TRUE ) )

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;

(a 'imp' (b 'imp' c)) . x = TRUE by A1, BVFUNC_1:def 11;
then ('not' (a . x)) 'or' ((b 'imp' c) . x) = TRUE by BVFUNC_1:def 8;
then A3: ('not' (a . x)) 'or' (('not' (b . x)) 'or' (c . x)) = TRUE by BVFUNC_1:def 8;
((a 'imp' b) 'imp' (a 'imp' c)) . x = ('not' ((a 'imp' b) . x)) 'or' ((a 'imp' c) . x) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' (b . x))) 'or' ((a 'imp' c) . x) by BVFUNC_1:def 8
.= (('not' ('not' (a . x))) '&' ('not' (b . x))) 'or' (('not' (a . x)) 'or' (c . x)) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' (c . x)) 'or' (a . x)) '&' ((('not' (a . x)) 'or' (c . x)) 'or' ('not' (b . x))) by XBOOLEAN:9
.= TRUE '&' ((('not' (a . x)) 'or' (c . x)) 'or' (a . x)) by A3, BINARITH:11
.= ((c . x) 'or' ('not' (a . x))) 'or' (a . x) by MARGREL1:14
.= (c . x) 'or' TRUE by A2, BINARITH:11
.= TRUE by BINARITH:10 ;
hence ((a 'imp' b) 'imp' (a 'imp' c)) . x = TRUE ; :: thesis: verum

end;

hence (a 'imp' b) 'imp' (a 'imp' c) = I_el Y by BVFUNC_1:def 11; :: thesis: verum