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

proof

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

end;

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