let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a '<' (a 'imp' b) 'eqv' b
let a, b be Function of Y,BOOLEAN; :: thesis: a '<' (a 'imp' b) 'eqv' b
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not a . z = TRUE or ((a 'imp' b) 'eqv' b) . z = TRUE )
assume a . z = TRUE ; :: thesis: ((a 'imp' b) 'eqv' b) . z = TRUE
then A2: 'not' (a . z) = FALSE ;
((a 'imp' b) 'eqv' b) . z = ((('not' a) 'or' b) 'eqv' b) . z by BVFUNC_4:8
.= (((('not' a) 'or' b) 'imp' b) '&' (b 'imp' (('not' a) 'or' b))) . z by BVFUNC_4:7
.= ((('not' (('not' a) 'or' b)) 'or' b) '&' (b 'imp' (('not' a) 'or' b))) . z by BVFUNC_4:8
.= ((('not' (('not' a) 'or' b)) 'or' b) '&' (('not' b) 'or' (('not' a) 'or' b))) . z by BVFUNC_4:8
.= ((('not' (('not' a) 'or' b)) 'or' b) . z) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 20
.= ((('not' (('not' a) 'or' b)) . z) 'or' (b . z)) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by BVFUNC_1:def 4
.= (('not' ((('not' a) 'or' b) . z)) 'or' (b . z)) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 19
.= (('not' ((('not' a) . z) 'or' (b . z))) 'or' (b . z)) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by BVFUNC_1:def 4
.= ((('not' ('not' (a . z))) '&' ('not' (b . z))) 'or' (b . z)) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 19
.= (((a . z) '&' ('not' (b . z))) 'or' (b . z)) '&' ((('not' b) . z) 'or' ((('not' a) 'or' b) . z)) by BVFUNC_1:def 4
.= (((a . z) '&' ('not' (b . z))) 'or' (b . z)) '&' ((('not' b) . z) 'or' ((('not' a) . z) 'or' (b . z))) by BVFUNC_1:def 4
.= (((a . z) '&' ('not' (b . z))) 'or' (b . z)) '&' ((('not' b) . z) 'or' (('not' (a . z)) 'or' (b . z))) by MARGREL1:def 19
.= ((TRUE '&' ('not' (b . z))) 'or' (b . z)) '&' (('not' (b . z)) 'or' (FALSE 'or' (b . z))) by A2, MARGREL1:def 19
.= (('not' (b . z)) 'or' (b . z)) '&' (('not' (b . z)) 'or' (FALSE 'or' (b . z)))
.= (('not' (b . z)) 'or' (b . z)) '&' (('not' (b . z)) 'or' (b . z))
.= TRUE by XBOOLEAN:102 ;
hence ((a 'imp' b) 'eqv' b) . z = TRUE ; :: thesis: verum