let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds
( a 'nand' (b 'nor' c) = (('not' a) 'or' b) 'or' c & a 'nand' (b 'nor' c) = a 'imp' (b 'or' c) )
let a, b, c be Function of Y,BOOLEAN; :: thesis: ( a 'nand' (b 'nor' c) = (('not' a) 'or' b) 'or' c & a 'nand' (b 'nor' c) = a 'imp' (b 'or' c) )
A1: a 'nand' (b 'nor' c) = 'not' (a '&' (b 'nor' c)) by th1
.= 'not' (a '&' ('not' (b 'or' c))) by Th2
.= ('not' a) 'or' ('not' ('not' (b 'or' c))) by BVFUNC_1:14
.= (('not' a) 'or' b) 'or' c by BVFUNC_1:8 ;
then a 'nand' (b 'nor' c) = ('not' a) 'or' (b 'or' c) by BVFUNC_1:8;
hence ( a 'nand' (b 'nor' c) = (('not' a) 'or' b) 'or' c & a 'nand' (b 'nor' c) = a 'imp' (b 'or' c) ) by A1, BVFUNC_4:8; :: thesis: verum