let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds ('not' a) 'imp' (('not' b) 'eqv' (b 'imp' a)) = I_el Y
let a, b be Function of Y,BOOLEAN; :: thesis: ('not' a) 'imp' (('not' b) 'eqv' (b 'imp' a)) = I_el Y
('not' a) 'imp' (('not' b) 'eqv' (b 'imp' a)) = ('not' ('not' a)) 'or' (('not' b) 'eqv' (b 'imp' a)) by BVFUNC_4:8
.= a 'or' ((('not' b) 'imp' (b 'imp' a)) '&' ((b 'imp' a) 'imp' ('not' b))) by BVFUNC_4:7
.= a 'or' ((('not' ('not' b)) 'or' (b 'imp' a)) '&' ((b 'imp' a) 'imp' ('not' b))) by BVFUNC_4:8
.= a 'or' ((b 'or' (('not' b) 'or' a)) '&' ((b 'imp' a) 'imp' ('not' b))) by BVFUNC_4:8
.= a 'or' (((b 'or' ('not' b)) 'or' a) '&' ((b 'imp' a) 'imp' ('not' b))) by BVFUNC_1:8
.= a 'or' (((I_el Y) 'or' a) '&' ((b 'imp' a) 'imp' ('not' b))) by BVFUNC_4:6
.= a 'or' ((I_el Y) '&' ((b 'imp' a) 'imp' ('not' b))) by BVFUNC_1:10
.= a 'or' ((b 'imp' a) 'imp' ('not' b)) by BVFUNC_1:6
.= a 'or' ((('not' b) 'or' a) 'imp' ('not' b)) by BVFUNC_4:8
.= a 'or' (('not' (('not' b) 'or' a)) 'or' ('not' b)) by BVFUNC_4:8
.= a 'or' ((('not' ('not' b)) '&' ('not' a)) 'or' ('not' b)) by BVFUNC_1:13
.= a 'or' ((b 'or' ('not' b)) '&' (('not' a) 'or' ('not' b))) by BVFUNC_1:11
.= a 'or' ((I_el Y) '&' (('not' a) 'or' ('not' b))) by BVFUNC_4:6
.= a 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:6
.= (a 'or' ('not' a)) 'or' ('not' b) by BVFUNC_1:8
.= (I_el Y) 'or' ('not' b) by BVFUNC_4:6 ;
hence ('not' a) 'imp' (('not' b) 'eqv' (b 'imp' a)) = I_el Y by BVFUNC_1:10; :: thesis: verum