let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds (a '&' (a 'imp' b)) '&' (b 'imp' c) '<' c
let a, b, c be Element of Funcs Y,BOOLEAN ; :: thesis: (a '&' (a 'imp' b)) '&' (b 'imp' c) '<' c
((a '&' (a 'imp' b)) '&' (b 'imp' c)) 'imp' c = ('not' ((a '&' (a 'imp' b)) '&' (b 'imp' c))) 'or' c by BVFUNC_4:8
.= ('not' ((a '&' b) '&' (b 'imp' c))) 'or' c by BVFUNC_7:10
.= ('not' (a '&' (b '&' (b 'imp' c)))) 'or' c by BVFUNC_1:7
.= ('not' (a '&' (b '&' c))) 'or' c by BVFUNC_7:10
.= ('not' ((a '&' b) '&' c)) 'or' c by BVFUNC_1:7
.= (('not' (a '&' b)) 'or' ('not' c)) 'or' c by BVFUNC_1:17
.= ('not' (a '&' b)) 'or' (('not' c) 'or' c) by BVFUNC_1:11
.= ('not' (a '&' b)) 'or' (I_el Y) by BVFUNC_4:6
.= I_el Y by BVFUNC_1:13 ;
hence (a '&' (a 'imp' b)) '&' (b 'imp' c) '<' c by BVFUNC_1:19; :: thesis: verum