let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' ('not' b) = ('not' a) '&' ('not' b)
let a, b be Function of Y,BOOLEAN; :: thesis: (a 'imp' b) '&' ('not' b) = ('not' a) '&' ('not' b)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((a 'imp' b) '&' ('not' b)) . x = (('not' a) '&' ('not' b)) . x
((a 'imp' b) '&' ('not' b)) . x = ((a 'imp' b) . x) '&' (('not' b) . x) by MARGREL1:def 20
.= (('not' b) . x) '&' (('not' (a . x)) 'or' (b . x)) by BVFUNC_1:def 8
.= ((('not' b) . x) '&' ('not' (a . x))) 'or' ((('not' b) . x) '&' (b . x)) by XBOOLEAN:8
.= ((('not' b) . x) '&' ('not' (a . x))) 'or' ((b . x) '&' ('not' (b . x))) by MARGREL1:def 19
.= ((('not' b) . x) '&' ('not' (a . x))) 'or' FALSE by XBOOLEAN:138
.= (('not' b) . x) '&' ('not' (a . x)) by BINARITH:3
.= (('not' b) . x) '&' (('not' a) . x) by MARGREL1:def 19
.= (('not' a) '&' ('not' b)) . x by MARGREL1:def 20 ;
hence ((a 'imp' b) '&' ('not' b)) . x = (('not' a) '&' ('not' b)) . x ; :: thesis: verum