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

proof

let x be Element of Y; :: thesis: (b 'imp' c) . x = TRUE
A2: (b 'imp' c) . x = ('not' (b . x)) 'or' (c . x) by BVFUNC_1:def 8;
(b 'imp' (b 'imp' c)) . x = TRUE by A1, BVFUNC_1:def 11;
then ('not' (b . x)) 'or' ((b 'imp' c) . x) = TRUE by BVFUNC_1:def 8;
then ('not' (b . x)) 'or' (('not' (b . x)) 'or' (c . x)) = TRUE by BVFUNC_1:def 8;
then A3: (('not' (b . x)) 'or' ('not' (b . x))) 'or' (c . x) = TRUE by BINARITH:11;
A4: ( 'not' (b . x) = TRUE or 'not' (b . x) = FALSE ) by XBOOLEAN:def 3;

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

per cases ( 'not' (b . x) = TRUE or c . x = TRUE ) by A3, A4, BINARITH:3;

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

hence (b 'imp' c) . x = TRUE by A2, BINARITH:10; :: thesis: verum

end;

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

hence (b 'imp' c) . x = TRUE by A2, BINARITH:10; :: thesis: verum

end;

end;

end;

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

end;

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