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

proof

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

end;

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