let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds
( a 'imp' b = I_el Y iff ('not' b) 'imp' ('not' a) = I_el Y )
let a, b be Function of Y,BOOLEAN; :: thesis: ( a 'imp' b = I_el Y iff ('not' b) 'imp' ('not' a) = I_el Y )

A1: now :: thesis: ( ('not' b) 'imp' ('not' a) = I_el Y implies a 'imp' b = I_el Y )

assume A2: ('not' b) 'imp' ('not' a) = I_el Y ; :: thesis: a 'imp' b = I_el Y
for x being Element of Y holds (a 'imp' b) . x = TRUE

proof

let x be Element of Y; :: thesis: (a 'imp' b) . x = TRUE
(('not' b) 'imp' ('not' a)) . x = TRUE by A2, BVFUNC_1:def 11;
then ('not' (('not' b) . x)) 'or' (('not' a) . x) = TRUE by BVFUNC_1:def 8;
then ('not' ('not' (b . x))) 'or' (('not' a) . x) = TRUE by MARGREL1:def 19;
then A3: ('not' ('not' (b . x))) 'or' ('not' (a . x)) = TRUE by MARGREL1:def 19;
A4: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;

now :: thesis: ( ( 'not' (a . x) = TRUE & (a 'imp' b) . x = TRUE ) or ( b . x = TRUE & (a 'imp' b) . x = TRUE ) )

per cases ( 'not' (a . x) = TRUE or b . x = TRUE ) by A3, A4, BINARITH:3;

case 'not' (a . x) = TRUE ; :: thesis: (a 'imp' b) . x = TRUE

then (a 'imp' b) . x = TRUE 'or' (b . x) by BVFUNC_1:def 8
.= TRUE by BINARITH:10 ;
hence (a 'imp' b) . x = TRUE ; :: thesis: verum

end;

case b . x = TRUE ; :: thesis: (a 'imp' b) . x = TRUE

then (a 'imp' b) . x = ('not' (a . x)) 'or' TRUE by BVFUNC_1:def 8
.= TRUE by BINARITH:10 ;
hence (a 'imp' b) . x = TRUE ; :: thesis: verum

end;

end;

end;

hence (a 'imp' b) . x = TRUE ; :: thesis: verum

end;

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

end;

now :: thesis: ( a 'imp' b = I_el Y implies ('not' b) 'imp' ('not' a) = I_el Y )

assume A5: a 'imp' b = I_el Y ; :: thesis: ('not' b) 'imp' ('not' a) = I_el Y
for x being Element of Y holds (('not' b) 'imp' ('not' a)) . x = TRUE

proof

let x be Element of Y; :: thesis: (('not' b) 'imp' ('not' a)) . x = TRUE
(a 'imp' b) . x = TRUE by A5, BVFUNC_1:def 11;
then A6: ('not' (a . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 8;
A7: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;

now :: thesis: ( ( 'not' (a . x) = TRUE & (('not' b) 'imp' ('not' a)) . x = TRUE ) or ( b . x = TRUE & (('not' b) 'imp' ('not' a)) . x = TRUE ) )

per cases ( 'not' (a . x) = TRUE or b . x = TRUE ) by A6, A7, BINARITH:3;

case A8: 'not' (a . x) = TRUE ; :: thesis: (('not' b) 'imp' ('not' a)) . x = TRUE

(('not' b) 'imp' ('not' a)) . x = ('not' (('not' b) . x)) 'or' (('not' a) . x) by BVFUNC_1:def 8
.= ('not' (('not' b) . x)) 'or' TRUE by A8, MARGREL1:def 19
.= TRUE by BINARITH:10 ;
hence (('not' b) 'imp' ('not' a)) . x = TRUE ; :: thesis: verum

end;

case A9: b . x = TRUE ; :: thesis: (('not' b) 'imp' ('not' a)) . x = TRUE

('not' b) . x = 'not' (b . x) by MARGREL1:def 19;
then (('not' b) 'imp' ('not' a)) . x = ('not' ('not' (b . x))) 'or' (('not' a) . x) by BVFUNC_1:def 8
.= TRUE by A9, BINARITH:10 ;
hence (('not' b) 'imp' ('not' a)) . x = TRUE ; :: thesis: verum

end;

end;

end;

hence (('not' b) 'imp' ('not' a)) . x = TRUE ; :: thesis: verum

end;

hence ('not' b) 'imp' ('not' a) = I_el Y by BVFUNC_1:def 11; :: thesis: verum

end;

hence ( a 'imp' b = I_el Y iff ('not' b) 'imp' ('not' a) = I_el Y ) by A1; :: thesis: verum