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

proof

let x be Element of Y; :: thesis: ((b 'imp' c) 'imp' (a 'imp' c)) . x = TRUE
A2: ((b 'imp' c) 'imp' (a 'imp' c)) . x = ('not' ((b 'imp' c) . x)) 'or' ((a 'imp' c) . x) by BVFUNC_1:def 8
.= ('not' (('not' (b . x)) 'or' (c . x))) 'or' ((a 'imp' c) . x) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' (c . x)) 'or' ((b . x) '&' ('not' (c . x))) by BVFUNC_1:def 8 ;

A3: now :: thesis: ( ( c . x = TRUE & ('not' (c . x)) 'or' (c . x) = TRUE ) or ( c . x = FALSE & ('not' (c . x)) 'or' (c . x) = TRUE ) )

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;

(a 'imp' b) . x = TRUE by A1, BVFUNC_1:def 11;
then A4: ('not' (a . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 8;
A5: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;

now :: thesis: ( ( 'not' (a . x) = TRUE & ((b 'imp' c) 'imp' (a 'imp' c)) . x = TRUE ) or ( b . x = TRUE & ((b 'imp' c) 'imp' (a 'imp' c)) . x = TRUE ) )

per cases ( 'not' (a . x) = TRUE or b . x = TRUE ) by A4, A5, BINARITH:3;

case 'not' (a . x) = TRUE ; :: thesis: ((b 'imp' c) 'imp' (a 'imp' c)) . x = TRUE

then ((b 'imp' c) 'imp' (a 'imp' c)) . x = TRUE 'or' ((b . x) '&' ('not' (c . x))) by A2, BINARITH:10
.= TRUE by BINARITH:10 ;
hence ((b 'imp' c) 'imp' (a 'imp' c)) . x = TRUE ; :: thesis: verum

end;

case b . x = TRUE ; :: thesis: ((b 'imp' c) 'imp' (a 'imp' c)) . x = TRUE

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

end;

end;

end;

hence ((b 'imp' c) 'imp' (a 'imp' c)) . x = TRUE ; :: thesis: verum

end;

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