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