let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a 'nand' b) 'nand' (a 'nand' c) = a '&' (b 'or' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a 'nand' b) 'nand' (a 'nand' c) = a '&' (b 'or' c)
thus (a 'nand' b) 'nand' (a 'nand' c) = 'not' ((a 'nand' b) '&' (a 'nand' c)) by th1
.= 'not' (('not' (a '&' b)) '&' (a 'nand' c)) by th1
.= 'not' (('not' (a '&' b)) '&' ('not' (a '&' c))) by th1
.= ('not' ('not' (a '&' b))) 'or' ('not' ('not' (a '&' c))) by BVFUNC_1:14
.= a '&' (b 'or' c) by BVFUNC_1:12 ; :: thesis: verum