let Y be non empty set ; :: thesis: for a, b, c, d being Function of Y,BOOLEAN holds a 'imp' ((b '&' c) '&' d) = ((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)
let a, b, c, d be Function of Y,BOOLEAN; :: thesis: a 'imp' ((b '&' c) '&' d) = ((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11((a 'imp' ((b '&' c) '&' d)),x) = K11((((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)),x)
(((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)) . x = (((a 'imp' b) '&' (a 'imp' c)) . x) '&' ((a 'imp' d) . x) by MARGREL1:def 20
.= (((a 'imp' b) . x) '&' ((a 'imp' c) . x)) '&' ((a 'imp' d) . x) by MARGREL1:def 20
.= ((('not' (a . x)) 'or' (b . x)) '&' ((a 'imp' c) . x)) '&' ((a 'imp' d) . x) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' (b . x)) '&' (('not' (a . x)) 'or' (c . x))) '&' ((a 'imp' d) . x) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' (b . x)) '&' (('not' (a . x)) 'or' (c . x))) '&' (('not' (a . x)) 'or' (d . x)) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' ((b . x) '&' (c . x))) '&' (('not' (a . x)) 'or' (d . x)) by XBOOLEAN:9
.= ('not' (a . x)) 'or' (((b . x) '&' (c . x)) '&' (d . x)) by XBOOLEAN:9
.= ('not' (a . x)) 'or' (((b '&' c) . x) '&' (d . x)) by MARGREL1:def 20
.= ('not' (a . x)) 'or' (((b '&' c) '&' d) . x) by MARGREL1:def 20
.= (a 'imp' ((b '&' c) '&' d)) . x by BVFUNC_1:def 8 ;
hence K11((a 'imp' ((b '&' c) '&' d)),x) = K11((((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)),x) ; :: thesis: verum