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