let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds
( a 'nand' (a 'nand' b) = ('not' a) 'or' b & a 'nand' (a 'nand' b) = a 'imp' b )
let a, b be Function of Y,BOOLEAN; :: thesis: ( a 'nand' (a 'nand' b) = ('not' a) 'or' b & a 'nand' (a 'nand' b) = a 'imp' b )
a 'nand' (a 'nand' b) = 'not' (a '&' (a 'nand' b)) by th1
.= 'not' (a '&' ('not' (a '&' b))) by th1
.= ('not' a) 'or' ('not' ('not' (a '&' b))) by BVFUNC_1:14
.= (('not' a) 'or' a) '&' (('not' a) 'or' b) by BVFUNC_1:11
.= (I_el Y) '&' (('not' a) 'or' b) by BVFUNC_4:6
.= ('not' a) 'or' b by BVFUNC_1:6 ;
hence ( a 'nand' (a 'nand' b) = ('not' a) 'or' b & a 'nand' (a 'nand' b) = a 'imp' b ) by BVFUNC_4:8; :: thesis: verum