let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds ((a '&' b) 'imp' c) 'imp' ((a '&' b) 'imp' (c '&' b)) = I_el Y
let a, b, c be Function of Y,BOOLEAN; :: thesis: ((a '&' b) 'imp' c) 'imp' ((a '&' b) 'imp' (c '&' b)) = I_el Y
((a '&' b) 'imp' c) 'imp' ((a '&' b) 'imp' (c '&' b)) = (('not' (a '&' b)) 'or' c) 'imp' ((a '&' b) 'imp' (c '&' b)) by BVFUNC_4:8
.= ('not' (('not' (a '&' b)) 'or' c)) 'or' ((a '&' b) 'imp' (c '&' b)) by BVFUNC_4:8
.= ('not' (('not' (a '&' b)) 'or' c)) 'or' (('not' (a '&' b)) 'or' (c '&' b)) by BVFUNC_4:8
.= (('not' ('not' (a '&' b))) '&' ('not' c)) 'or' (('not' (a '&' b)) 'or' (c '&' b)) by BVFUNC_1:13
.= ((a '&' b) '&' ('not' c)) 'or' ((('not' a) 'or' ('not' b)) 'or' (c '&' b)) by BVFUNC_1:14
.= ((a '&' b) '&' ('not' c)) 'or' (('not' a) 'or' (('not' b) 'or' (b '&' c))) by BVFUNC_1:8
.= ((a '&' b) '&' ('not' c)) 'or' (('not' a) 'or' ((('not' b) 'or' b) '&' (('not' b) 'or' c))) by BVFUNC_1:11
.= ((a '&' b) '&' ('not' c)) 'or' (('not' a) 'or' ((I_el Y) '&' (('not' b) 'or' c))) by BVFUNC_4:6
.= ((a '&' b) '&' ('not' c)) 'or' (('not' a) 'or' (('not' b) 'or' c)) by BVFUNC_1:6
.= (((a '&' b) '&' ('not' c)) 'or' ('not' a)) 'or' (('not' b) 'or' c) by BVFUNC_1:8
.= (((a '&' b) 'or' ('not' a)) '&' (('not' c) 'or' ('not' a))) 'or' (('not' b) 'or' c) by BVFUNC_1:11
.= (((a 'or' ('not' a)) '&' (b 'or' ('not' a))) '&' (('not' c) 'or' ('not' a))) 'or' (('not' b) 'or' c) by BVFUNC_1:11
.= (((I_el Y) '&' (b 'or' ('not' a))) '&' (('not' c) 'or' ('not' a))) 'or' (('not' b) 'or' c) by BVFUNC_4:6
.= ((b 'or' ('not' a)) '&' (('not' c) 'or' ('not' a))) 'or' (('not' b) 'or' c) by BVFUNC_1:6
.= (((b 'or' ('not' a)) '&' (('not' c) 'or' ('not' a))) 'or' c) 'or' ('not' b) by BVFUNC_1:8
.= (((b 'or' ('not' a)) 'or' c) '&' ((('not' c) 'or' ('not' a)) 'or' c)) 'or' ('not' b) by BVFUNC_1:11
.= (((b 'or' ('not' a)) 'or' c) '&' (('not' a) 'or' (('not' c) 'or' c))) 'or' ('not' b) by BVFUNC_1:8
.= (((b 'or' ('not' a)) 'or' c) '&' (('not' a) 'or' (I_el Y))) 'or' ('not' b) by BVFUNC_4:6
.= (((b 'or' ('not' a)) 'or' c) '&' (I_el Y)) 'or' ('not' b) by BVFUNC_1:10
.= ((b 'or' ('not' a)) 'or' c) 'or' ('not' b) by BVFUNC_1:6
.= (('not' b) 'or' (b 'or' ('not' a))) 'or' c by BVFUNC_1:8
.= ((('not' b) 'or' b) 'or' ('not' a)) 'or' c by BVFUNC_1:8
.= ((I_el Y) 'or' ('not' a)) 'or' c by BVFUNC_4:6
.= (I_el Y) 'or' c by BVFUNC_1:10 ;
hence ((a '&' b) 'imp' c) 'imp' ((a '&' b) 'imp' (c '&' b)) = I_el Y by BVFUNC_1:10; :: thesis: verum