let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)) = I_el Y
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)) = I_el Y
for x being Element of Y holds ((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))) . x = TRUE

proof

let x be Element of Y; :: thesis: ((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))) . x = TRUE
((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))) . x = ('not' ((a 'imp' c) . x)) 'or' (((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)) . x) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' (((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)) . x) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' ((b 'imp' c) . x)) 'or' (((a 'or' b) 'imp' c) . x)) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (('not' (b . x)) 'or' (c . x))) 'or' (((a 'or' b) 'imp' c) . x)) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' ((a 'or' b) . x)) 'or' (c . x))) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' ((a . x) 'or' (b . x))) 'or' (c . x))) by BVFUNC_1:def 4
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (('not' (b . x)) 'or' (c . x))) 'or' (((c . x) 'or' ('not' (a . x))) '&' (('not' (b . x)) 'or' (c . x)))) by XBOOLEAN:9
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' ((('not' (('not' (b . x)) 'or' (c . x))) 'or' ((c . x) 'or' ('not' (a . x)))) '&' (('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' (b . x)) 'or' (c . x)))) by XBOOLEAN:9
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' (TRUE '&' (('not' (('not' (b . x)) 'or' (c . x))) 'or' ((c . x) 'or' ('not' (a . x))))) by XBOOLEAN:102
.= ('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' (a . x)) 'or' (c . x)))
.= (('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' ('not' (('not' (b . x)) 'or' (c . x)))
.= TRUE 'or' ('not' (('not' (b . x)) 'or' (c . x))) by XBOOLEAN:102
.= TRUE ;
hence ((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))) . x = TRUE ; :: thesis: verum

end;

hence (a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)) = I_el Y by BVFUNC_1:def 11; :: thesis: verum