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