let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a 'xor' b = 'not' (a 'eqv' b)
let a, b be Function of Y,BOOLEAN; :: thesis: a 'xor' b = 'not' (a 'eqv' b)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11((a 'xor' b),x) = K11(('not' (a 'eqv' b)),x)
(a 'xor' b) . x = ('not' ('not' ((('not' a) '&' b) 'or' (a '&' ('not' b))))) . x by BVFUNC_4:9
.= ('not' (('not' (('not' a) '&' b)) '&' ('not' (a '&' ('not' b))))) . x by BVFUNC_1:13
.= ('not' ((('not' ('not' a)) 'or' ('not' b)) '&' ('not' (a '&' ('not' b))))) . x by BVFUNC_1:14
.= ('not' ((a 'or' ('not' b)) '&' (('not' a) 'or' ('not' ('not' b))))) . x by BVFUNC_1:14
.= ('not' ((b 'imp' a) '&' (('not' a) 'or' b))) . x by BVFUNC_4:8
.= ('not' ((b 'imp' a) '&' (a 'imp' b))) . x by BVFUNC_4:8
.= ('not' (a 'eqv' b)) . x by BVFUNC_4:7 ;
hence K11((a 'xor' b),x) = K11(('not' (a 'eqv' b)),x) ; :: thesis: verum