let Y be non empty set ; :: thesis: for a, b, c, d being Function of Y,BOOLEAN holds ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)
let a, b, c, d be Function of Y,BOOLEAN; :: thesis: ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11((((a '&' b) '&' c) 'imp' d),x) = K11((((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)),x)
(((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x = (((a 'imp' d) 'or' (b 'imp' d)) . x) 'or' ((c 'imp' d) . x) by BVFUNC_1:def 4
.= (((a 'imp' d) . x) 'or' ((b 'imp' d) . x)) 'or' ((c 'imp' d) . x) by BVFUNC_1:def 4
.= ((('not' (a . x)) 'or' (d . x)) 'or' ((b 'imp' d) . x)) 'or' ((c 'imp' d) . x) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' (d . x)) 'or' (('not' (b . x)) 'or' (d . x))) 'or' ((c 'imp' d) . x) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' (d . x)) 'or' (('not' (b . x)) 'or' (d . x))) 'or' (('not' (c . x)) 'or' (d . x)) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' ((d . x) 'or' ('not' (b . x)))) 'or' (d . x)) 'or' (('not' (c . x)) 'or' (d . x))
.= (((('not' (a . x)) 'or' ('not' (b . x))) 'or' (d . x)) 'or' (d . x)) 'or' (('not' (c . x)) 'or' (d . x))
.= ((('not' (a . x)) 'or' ('not' (b . x))) 'or' ((d . x) 'or' (d . x))) 'or' (('not' (c . x)) 'or' (d . x)) by BINARITH:11
.= (('not' ((a . x) '&' (b . x))) 'or' ((d . x) 'or' ('not' (c . x)))) 'or' (d . x)
.= ((('not' ((a . x) '&' (b . x))) 'or' ('not' (c . x))) 'or' (d . x)) 'or' (d . x)
.= (('not' ((a . x) '&' (b . x))) 'or' ('not' (c . x))) 'or' ((d . x) 'or' (d . x)) by BINARITH:11
.= ('not' (((a '&' b) . x) '&' (c . x))) 'or' (d . x) by MARGREL1:def 20
.= ('not' (((a '&' b) '&' c) . x)) 'or' (d . x) by MARGREL1:def 20
.= (((a '&' b) '&' c) 'imp' d) . x by BVFUNC_1:def 8 ;
hence K11((((a '&' b) '&' c) 'imp' d),x) = K11((((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)),x) ; :: thesis: verum