let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c = a 'imp' (b 'imp' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a '&' b) 'imp' c = a 'imp' (b 'imp' c)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((a '&' b) 'imp' c) . x = (a 'imp' (b 'imp' c)) . x
(a 'imp' (b 'imp' c)) . x = ('not' (a . x)) 'or' ((b 'imp' c) . x) by BVFUNC_1:def 8
.= ('not' (a . x)) 'or' (('not' (b . x)) 'or' (c . x)) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' ('not' (b . x))) 'or' (c . x) by BINARITH:11
.= ('not' ((a '&' b) . x)) 'or' (c . x) by MARGREL1:def 20
.= ((a '&' b) 'imp' c) . x by BVFUNC_1:def 8 ;
hence ((a '&' b) 'imp' c) . x = (a 'imp' (b 'imp' c)) . x ; :: thesis: verum