let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds (a 'nand' b) 'nand' (a 'nand' c) = a '&' (b 'or' c)
let a, b, c be Element of Funcs 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_1:17
.= a '&' (b 'or' c) by BVFUNC_1:15 ; :: thesis: verum