let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c) = (a '&' b) '&' c
let a, b, c be Function of Y,BOOLEAN; :: thesis: (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c) = (a '&' b) '&' c
(((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c) = (((('not' a) 'or' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c) by BVFUNC_4:8
.= (((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c) by BVFUNC_4:8
.= (((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (('not' c) 'or' a)) '&' ((a 'or' b) 'or' c) by BVFUNC_4:8
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' ((('not' c) 'or' a) '&' ((a 'or' b) 'or' c)) by BVFUNC_1:4
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (((('not' c) 'or' a) '&' (a 'or' b)) 'or' ((('not' c) 'or' a) '&' c)) by BVFUNC_1:12
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (((('not' c) 'or' a) '&' (a 'or' b)) 'or' ((('not' c) '&' c) 'or' (a '&' c))) by BVFUNC_1:12
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (((('not' c) 'or' a) '&' (a 'or' b)) 'or' ((O_el Y) 'or' (a '&' c))) by BVFUNC_4:5
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (((('not' c) 'or' a) '&' (a 'or' b)) 'or' (a '&' c)) by BVFUNC_1:9
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' ((a 'or' (('not' c) '&' b)) 'or' (a '&' c)) by BVFUNC_1:11
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' ((a 'or' (a '&' c)) 'or' (('not' c) '&' b)) by BVFUNC_1:8
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (((a '&' (I_el Y)) 'or' (a '&' c)) 'or' (('not' c) '&' b)) by BVFUNC_1:6
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' ((a '&' ((I_el Y) 'or' c)) 'or' (('not' c) '&' b)) by BVFUNC_1:12
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' ((a '&' (I_el Y)) 'or' (('not' c) '&' b)) by BVFUNC_1:10
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (a 'or' (('not' c) '&' b)) by BVFUNC_1:6
.= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' ((a 'or' ('not' c)) '&' (a 'or' b)) by BVFUNC_1:11
.= ((a 'or' b) '&' ((('not' a) 'or' b) '&' (('not' b) 'or' c))) '&' (a 'or' ('not' c)) by BVFUNC_1:4
.= (((a 'or' b) '&' (('not' a) 'or' b)) '&' (('not' b) 'or' c)) '&' (a 'or' ('not' c)) by BVFUNC_1:4
.= (((a '&' ('not' a)) 'or' b) '&' (('not' b) 'or' c)) '&' (a 'or' ('not' c)) by BVFUNC_1:11
.= (((O_el Y) 'or' b) '&' (('not' b) 'or' c)) '&' (a 'or' ('not' c)) by BVFUNC_4:5
.= (b '&' (('not' b) 'or' c)) '&' (a 'or' ('not' c)) by BVFUNC_1:9
.= ((b '&' ('not' b)) 'or' (b '&' c)) '&' (a 'or' ('not' c)) by BVFUNC_1:12
.= ((O_el Y) 'or' (b '&' c)) '&' (a 'or' ('not' c)) by BVFUNC_4:5
.= (b '&' c) '&' (a 'or' ('not' c)) by BVFUNC_1:9
.= ((b '&' c) '&' a) 'or' ((b '&' c) '&' ('not' c)) by BVFUNC_1:12
.= ((b '&' c) '&' a) 'or' (b '&' (c '&' ('not' c))) by BVFUNC_1:4
.= ((b '&' c) '&' a) 'or' (b '&' (O_el Y)) by BVFUNC_4:5
.= ((b '&' c) '&' a) 'or' (O_el Y) by BVFUNC_1:5
.= (b '&' c) '&' a by BVFUNC_1:9
.= (a '&' b) '&' c by BVFUNC_1:4 ;
hence (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c) = (a '&' b) '&' c ; :: thesis: verum