let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN st (a '&' ('not' b)) 'imp' ('not' a) = I_el Y holds
a 'imp' b = I_el Y
let a, b be Function of Y,BOOLEAN; :: thesis: ( (a '&' ('not' b)) 'imp' ('not' a) = I_el Y implies a 'imp' b = I_el Y )
assume A1: (a '&' ('not' b)) 'imp' ('not' a) = I_el Y ; :: thesis: a 'imp' b = I_el Y
for x being Element of Y holds (a 'imp' b) . x = TRUE

proof

let x be Element of Y; :: thesis: (a 'imp' b) . x = TRUE
((a '&' ('not' b)) 'imp' ('not' a)) . x = TRUE by A1, BVFUNC_1:def 11;
then ('not' ((a '&' ('not' b)) . x)) 'or' (('not' a) . x) = TRUE by BVFUNC_1:def 8;
then ('not' ((a . x) '&' (('not' b) . x))) 'or' (('not' a) . x) = TRUE by MARGREL1:def 20;
then (('not' (a . x)) 'or' ('not' ('not' (b . x)))) 'or' (('not' a) . x) = TRUE by MARGREL1:def 19;
then (('not' (a . x)) 'or' (b . x)) 'or' ('not' (a . x)) = TRUE by MARGREL1:def 19;
then (b . x) 'or' (('not' (a . x)) 'or' ('not' (a . x))) = TRUE by XBOOLEAN:4;
hence (a 'imp' b) . x = TRUE by BVFUNC_1:def 8; :: thesis: verum

end;

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