let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a '&' b) 'or' (('not' a) '&' ('not' b))
let a, b be Function of Y,BOOLEAN; :: thesis: a 'eqv' b = (a '&' b) 'or' (('not' a) '&' ('not' b))
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11((a 'eqv' b),x) = K11(((a '&' b) 'or' (('not' a) '&' ('not' b))),x)
((a '&' b) 'or' (('not' a) '&' ('not' b))) . x = (((a '&' b) 'or' ('not' a)) '&' ((a '&' b) 'or' ('not' b))) . x by BVFUNC_1:11
.= (((a 'or' ('not' a)) '&' (b 'or' ('not' a))) '&' ((a '&' b) 'or' ('not' b))) . x by BVFUNC_1:11
.= (((a 'or' ('not' a)) '&' (b 'or' ('not' a))) '&' ((a 'or' ('not' b)) '&' (b 'or' ('not' b)))) . x by BVFUNC_1:11
.= (((I_el Y) '&' (b 'or' ('not' a))) '&' ((a 'or' ('not' b)) '&' (b 'or' ('not' b)))) . x by BVFUNC_4:6
.= (((I_el Y) '&' (b 'or' ('not' a))) '&' ((a 'or' ('not' b)) '&' (I_el Y))) . x by BVFUNC_4:6
.= ((b 'or' ('not' a)) '&' ((a 'or' ('not' b)) '&' (I_el Y))) . x by BVFUNC_1:6
.= ((b 'or' ('not' a)) '&' (a 'or' ('not' b))) . x by BVFUNC_1:6
.= (a 'eqv' b) . x by Th18 ;
hence K11((a 'eqv' b),x) = K11(((a '&' b) 'or' (('not' a) '&' ('not' b))),x) ; :: thesis: verum