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

proof

let x be Element of Y; :: thesis: (a 'imp' (('not' a) 'eqv' ('not' a))) . x = TRUE
(a 'imp' (('not' a) 'eqv' ('not' a))) . x = (('not' a) 'or' (('not' a) 'eqv' ('not' a))) . x by BVFUNC_4:8
.= (('not' a) 'or' ((('not' a) 'imp' ('not' a)) '&' (('not' a) 'imp' ('not' a)))) . x by BVFUNC_4:7
.= (('not' a) 'or' (('not' ('not' a)) 'or' ('not' a))) . x by BVFUNC_4:8
.= (('not' a) 'or' (I_el Y)) . x by BVFUNC_4:6
.= TRUE by BVFUNC_1:10, BVFUNC_1:def 11 ;
hence (a 'imp' (('not' a) 'eqv' ('not' a))) . x = TRUE ; :: thesis: verum

end;

hence a 'imp' (('not' a) 'eqv' ('not' a)) = I_el Y by BVFUNC_1:def 11; :: thesis: verum