let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN st 'not' a = I_el Y holds
a 'imp' b = I_el Y
let a, b be Function of Y,BOOLEAN; :: thesis: ( 'not' a = I_el Y implies a 'imp' b = I_el Y )
assume A1: '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
('not' a) . x = TRUE by A1, BVFUNC_1:def 11;
then 'not' (a . x) = TRUE by MARGREL1:def 19;
then (a 'imp' b) . x = TRUE 'or' (b . x) by BVFUNC_1:def 8
.= TRUE by BINARITH:10 ;
hence (a 'imp' b) . x = TRUE ; :: thesis: verum

end;

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