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

proof

let x be Element of Y; :: thesis: (((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)) . x = TRUE
(((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)) . x = ('not' (((a 'imp' c) '&' (b 'imp' c)) . x)) 'or' (((a 'or' b) 'imp' c) . x) by BVFUNC_1:def 8
.= ('not' (((a 'imp' c) . x) '&' ((b 'imp' c) . x))) 'or' (((a 'or' b) 'imp' c) . x) by MARGREL1:def 20
.= ('not' ((('not' (a . x)) 'or' (c . x)) '&' ((b 'imp' c) . x))) 'or' (((a 'or' b) 'imp' c) . x) by BVFUNC_1:def 8
.= ('not' ((('not' (a . x)) 'or' (c . x)) '&' (('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)) '&' (('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' ((c . x) 'or' ('not' ((a . x) 'or' (b . x)))) by BVFUNC_1:def 4
.= (('not' (('not' (a . x)) 'or' (c . x))) 'or' ('not' (('not' (b . x)) 'or' (c . x)))) 'or' ((('not' (a . x)) 'or' (c . x)) '&' ((c . x) 'or' ('not' (b . x)))) by XBOOLEAN:9
.= ((('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' (('not' (b . x)) 'or' (c . x)))) 'or' (('not' (b . x)) 'or' (c . x))) by XBOOLEAN:9
.= (('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (a . x)) 'or' (c . x)))) '&' (('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' (b . x)) 'or' (c . x))))
.= (('not' (('not' (b . x)) 'or' (c . x))) 'or' TRUE) '&' (('not' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (('not' (b . x)) 'or' (c . x))) 'or' (('not' (b . x)) 'or' (c . x)))) by XBOOLEAN:102
.= TRUE '&' (('not' (('not' (a . x)) 'or' (c . x))) 'or' TRUE) by XBOOLEAN:102
.= TRUE ;
hence (((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)) . x = TRUE ; :: thesis: verum

end;

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