let Y be non empty set ; :: thesis: for a, b, c, d being Function of Y,BOOLEAN holds (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = I_el Y
let a, b, c, d be Function of Y,BOOLEAN; :: thesis: (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = I_el Y
for x being Element of Y holds ((((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b))) . x = TRUE

proof

let x be Element of Y; :: thesis: ((((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b))) . x = TRUE
(((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = ('not' (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_4:8
.= ('not' (((('not' a) 'or' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_4:8
.= ('not' (((('not' a) 'or' c) '&' (('not' b) 'or' d)) '&' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_4:8
.= (('not' ((('not' a) 'or' c) '&' (('not' b) 'or' d))) 'or' ('not' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:14
.= ((('not' (('not' a) 'or' c)) 'or' ('not' (('not' b) 'or' d))) 'or' ('not' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:14
.= (((('not' ('not' a)) '&' ('not' c)) 'or' ('not' (('not' b) 'or' d))) 'or' ('not' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:13
.= (((a '&' ('not' c)) 'or' (('not' ('not' b)) '&' ('not' d))) 'or' ('not' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:13
.= (((a '&' ('not' c)) 'or' (b '&' ('not' d))) 'or' (('not' ('not' c)) '&' ('not' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:13
.= ((a '&' ('not' c)) 'or' ((b '&' ('not' d)) 'or' (c '&' d))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:8
.= ((a '&' ('not' c)) 'or' ((b 'or' (c '&' d)) '&' (('not' d) 'or' (c '&' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:11
.= ((a '&' ('not' c)) 'or' ((b 'or' (c '&' d)) '&' ((('not' d) 'or' c) '&' (('not' d) 'or' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:11
.= ((a '&' ('not' c)) 'or' ((b 'or' (c '&' d)) '&' ((('not' d) 'or' c) '&' (I_el Y)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_4:6
.= ((a '&' ('not' c)) 'or' ((b 'or' (c '&' d)) '&' (('not' d) 'or' c))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:6
.= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' ((a '&' ('not' c)) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:8
.= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' ((a 'or' (('not' a) 'or' ('not' b))) '&' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:11
.= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' (((a 'or' ('not' a)) 'or' ('not' b)) '&' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:8
.= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' (((I_el Y) 'or' ('not' b)) '&' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_4:6
.= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' ((I_el Y) '&' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:10
.= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:6
.= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' ((('not' d) 'or' c) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:11
.= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' (((('not' d) 'or' c) 'or' ('not' c)) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:8
.= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' ((('not' d) 'or' (c 'or' ('not' c))) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:8
.= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' ((('not' d) 'or' (I_el Y)) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_4:6
.= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' ((I_el Y) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:10
.= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' (I_el Y) by BVFUNC_1:10
.= (b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:6
.= (c '&' d) 'or' (b 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:8
.= (c '&' d) 'or' ((b 'or' (('not' b) 'or' ('not' a))) 'or' ('not' c)) by BVFUNC_1:8
.= (c '&' d) 'or' (((b 'or' ('not' b)) 'or' ('not' a)) 'or' ('not' c)) by BVFUNC_1:8
.= (c '&' d) 'or' (((I_el Y) 'or' ('not' a)) 'or' ('not' c)) by BVFUNC_4:6
.= (c '&' d) 'or' ((I_el Y) 'or' ('not' c)) by BVFUNC_1:10
.= (c '&' d) 'or' (I_el Y) by BVFUNC_1:10
.= I_el Y by BVFUNC_1:10 ;
hence ((((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b))) . x = TRUE by BVFUNC_1:def 11; :: thesis: verum

end;

hence (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = I_el Y by BVFUNC_1:def 11; :: thesis: verum