let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds 'not' (a 'eqv' b) = a 'eqv' ('not' b)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: 'not' (a 'eqv' b) = a 'eqv' ('not' b)
A1: for x being Element of Y holds ('not' (a 'eqv' b)) . x = (a 'eqv' ('not' b)) . x

proof

let x be Element of Y; :: thesis: ('not' (a 'eqv' b)) . x = (a 'eqv' ('not' b)) . x
('not' (a 'eqv' b)) . x = ('not' ((a 'imp' b) '&' (b 'imp' a))) . x by BVFUNC_4:7
.= ('not' ((('not' a) 'or' b) '&' (b 'imp' a))) . x by BVFUNC_4:8
.= ('not' ((('not' a) 'or' b) '&' (('not' b) 'or' a))) . x by BVFUNC_4:8
.= (('not' (('not' a) 'or' b)) 'or' ('not' (('not' b) 'or' a))) . x by BVFUNC_1:17
.= ((('not' ('not' a)) '&' ('not' b)) 'or' ('not' (('not' b) 'or' a))) . x by BVFUNC_1:16
.= ((a '&' ('not' b)) 'or' (('not' ('not' b)) '&' ('not' a))) . x by BVFUNC_1:16
.= (((a '&' ('not' b)) 'or' b) '&' ((a '&' ('not' b)) 'or' ('not' a))) . x by BVFUNC_1:14
.= (((a 'or' b) '&' (('not' b) 'or' b)) '&' ((a '&' ('not' b)) 'or' ('not' a))) . x by BVFUNC_1:14
.= (((a 'or' b) '&' (('not' b) 'or' b)) '&' ((a 'or' ('not' a)) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_1:14
.= (((a 'or' b) '&' (I_el Y)) '&' ((a 'or' ('not' a)) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_4:6
.= (((a 'or' b) '&' (I_el Y)) '&' ((I_el Y) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_4:6
.= ((a 'or' b) '&' ((I_el Y) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_1:9
.= ((('not' a) 'or' ('not' b)) '&' (('not' ('not' b)) 'or' a)) . x by BVFUNC_1:9
.= ((('not' a) 'or' ('not' b)) '&' (('not' b) 'imp' a)) . x by BVFUNC_4:8
.= ((a 'imp' ('not' b)) '&' (('not' b) 'imp' a)) . x by BVFUNC_4:8
.= (a 'eqv' ('not' b)) . x by BVFUNC_4:7 ;
hence ('not' (a 'eqv' b)) . x = (a 'eqv' ('not' b)) . x ; :: thesis: verum

end;

consider k3 being Function such that
A2: ( 'not' (a 'eqv' b) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A3: ( a 'eqv' ('not' b) = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
( Y = dom k3 & Y = dom k4 & ( for u being set st u in Y holds
k3 . u = k4 . u ) ) by A1, A2, A3;
hence 'not' (a 'eqv' b) = a 'eqv' ('not' b) by A2, A3, FUNCT_1:9; :: thesis: verum