let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds 'not' (a 'eqv' b) = a 'eqv' ('not' b)
let a, b be Function of Y,BOOLEAN; :: thesis: 'not' (a 'eqv' b) = a 'eqv' ('not' b)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11(('not' (a 'eqv' b)),x) = K11((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_1:14
.= ((('not' ('not' a)) '&' ('not' b)) 'or' ('not' (('not' b) 'or' a))) . x by BVFUNC_1:13
.= ((a '&' ('not' b)) 'or' (('not' ('not' b)) '&' ('not' a))) . x by BVFUNC_1:13
.= (((a '&' ('not' b)) 'or' b) '&' ((a '&' ('not' b)) 'or' ('not' a))) . x by BVFUNC_1:11
.= (((a 'or' b) '&' (('not' b) 'or' b)) '&' ((a '&' ('not' b)) 'or' ('not' a))) . x by BVFUNC_1:11
.= (((a 'or' b) '&' (('not' b) 'or' b)) '&' ((a 'or' ('not' a)) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_1:11
.= (((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_1:6
.= ((('not' a) 'or' ('not' b)) '&' (('not' ('not' b)) 'or' a)) . x by BVFUNC_1:6
.= ((('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 K11(('not' (a 'eqv' b)),x) = K11((a 'eqv' ('not' b)),x) ; :: thesis: verum