let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'eqv' b '<' (a 'imp' c) 'eqv' (b 'imp' c)
let a, b, c be Element of Funcs (Y,BOOLEAN); :: thesis: a 'eqv' b '<' (a 'imp' c) 'eqv' (b 'imp' c)
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (a 'eqv' b) . z = TRUE or ((a 'imp' c) 'eqv' (b 'imp' c)) . z = TRUE )
A1: (a 'eqv' b) . z = ((a 'imp' b) '&' (b 'imp' a)) . z by BVFUNC_4:7
.= ((a 'imp' b) . z) '&' ((b 'imp' a) . z) by MARGREL1:def 21 ;
assume A2: (a 'eqv' b) . z = TRUE ; :: thesis: ((a 'imp' c) 'eqv' (b 'imp' c)) . z = TRUE
then (a 'imp' b) . z = TRUE by A1, MARGREL1:45;
then A3: ('not' (a . z)) 'or' (b . z) = TRUE by BVFUNC_1:def 11;
(b 'imp' a) . z = TRUE by A2, A1, MARGREL1:45;
then A4: ('not' (b . z)) 'or' (a . z) = TRUE by BVFUNC_1:def 11;
A5: ((a 'imp' c) 'eqv' (b 'imp' c)) . z = (((a 'imp' c) 'imp' (b 'imp' c)) '&' ((b 'imp' c) 'imp' (a 'imp' c))) . z by BVFUNC_4:7
.= (((a 'imp' c) 'imp' (b 'imp' c)) . z) '&' (((b 'imp' c) 'imp' (a 'imp' c)) . z) by MARGREL1:def 21
.= (('not' ((a 'imp' c) . z)) 'or' ((b 'imp' c) . z)) '&' (((b 'imp' c) 'imp' (a 'imp' c)) . z) by BVFUNC_1:def 11
.= (('not' ((a 'imp' c) . z)) 'or' ((b 'imp' c) . z)) '&' (('not' ((b 'imp' c) . z)) 'or' ((a 'imp' c) . z)) by BVFUNC_1:def 11
.= (('not' (('not' (a . z)) 'or' (c . z))) 'or' ((b 'imp' c) . z)) '&' (('not' ((b 'imp' c) . z)) 'or' ((a 'imp' c) . z)) by BVFUNC_1:def 11
.= (('not' (('not' (a . z)) 'or' (c . z))) 'or' (('not' (b . z)) 'or' (c . z))) '&' (('not' ((b 'imp' c) . z)) 'or' ((a 'imp' c) . z)) by BVFUNC_1:def 11
.= (('not' (('not' (a . z)) 'or' (c . z))) 'or' (('not' (b . z)) 'or' (c . z))) '&' (('not' (('not' (b . z)) 'or' (c . z))) 'or' ((a 'imp' c) . z)) by BVFUNC_1:def 11
.= (((a . z) '&' ('not' (c . z))) 'or' (('not' (b . z)) 'or' (c . z))) '&' (((b . z) '&' ('not' (c . z))) 'or' (('not' (a . z)) 'or' (c . z))) by BVFUNC_1:def 11 ;
A6: ( a . z = TRUE or a . z = FALSE ) by XBOOLEAN:def 3;
A7: (b 'imp' a) . z = ('not' (b . z)) 'or' (a . z) by BVFUNC_1:def 11;
A8: ( b . z = TRUE or b . z = FALSE ) by XBOOLEAN:def 3;

now

per cases ( 'not' (a . z) = TRUE or b . z = TRUE ) by A3, A8, BINARITH:7;

case A9: 'not' (a . z) = TRUE ; :: thesis: ((a 'imp' c) 'eqv' (b 'imp' c)) . z = TRUE

then A10: a . z = FALSE by MARGREL1:41;
then 'not' (b . z) = TRUE by A4, BINARITH:7;
then ((a 'imp' c) 'eqv' (b 'imp' c)) . z = (FALSE 'or' (TRUE 'or' (c . z))) '&' (FALSE 'or' (TRUE 'or' (c . z))) by A5, A9, A10, MARGREL1:49
.= (TRUE 'or' (c . z)) '&' (TRUE 'or' (c . z)) by BINARITH:7
.= TRUE by BINARITH:19 ;
hence ((a 'imp' c) 'eqv' (b 'imp' c)) . z = TRUE ; :: thesis: verum

end;

case A11: b . z = TRUE ; :: thesis: ((a 'imp' c) 'eqv' (b 'imp' c)) . z = TRUE

then 'not' (b . z) = FALSE by MARGREL1:41;
then ((a 'imp' c) 'eqv' (b 'imp' c)) . z = (('not' (c . z)) 'or' (FALSE 'or' (c . z))) '&' (('not' (c . z)) 'or' (FALSE 'or' (c . z))) by A2, A1, A7, A6, A5, A11, BINARITH:7, MARGREL1:45
.= (('not' (c . z)) 'or' (c . z)) '&' (('not' (c . z)) 'or' (c . z)) by BINARITH:7
.= TRUE by XBOOLEAN:102 ;
hence ((a 'imp' c) 'eqv' (b 'imp' c)) . z = TRUE ; :: thesis: verum

end;

end;

end;

hence ((a 'imp' c) 'eqv' (b 'imp' c)) . z = TRUE ; :: thesis: verum