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

A1: now

assume A2: 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 A2, BVFUNC_1:def 14;
then A3: ('not' (a . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 11;
A4: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;

now

per cases ( 'not' (a . x) = TRUE or b . x = TRUE ) by A3, A4, BINARITH:7;

case A5: '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 11
.= ('not' (('not' b) . x)) 'or' TRUE by A5, MARGREL1:def 20
.= TRUE by BINARITH:19 ;
hence (('not' b) 'imp' ('not' a)) . x = TRUE ; :: thesis: verum

end;

case A6: b . x = TRUE ; :: thesis: (('not' b) 'imp' ('not' a)) . x = TRUE

('not' b) . x = 'not' (b . x) by MARGREL1:def 20;
then (('not' b) 'imp' ('not' a)) . x = ('not' ('not' (b . x))) 'or' (('not' a) . x) by BVFUNC_1:def 11
.= TRUE by A6, BINARITH:19 ;
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 14; :: thesis: verum

end;

now

assume A7: ('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 A7, BVFUNC_1:def 14;
then ('not' (('not' b) . x)) 'or' (('not' a) . x) = TRUE by BVFUNC_1:def 11;
then ('not' ('not' (b . x))) 'or' (('not' a) . x) = TRUE by MARGREL1:def 20;
then A8: ('not' ('not' (b . x))) 'or' ('not' (a . x)) = TRUE by MARGREL1:def 20;
A9: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;

now

per cases ( 'not' (a . x) = TRUE or b . x = TRUE ) by A8, A9, BINARITH:7;

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

then (a 'imp' b) . x = TRUE 'or' (b . x) by BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
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 11
.= TRUE by BINARITH:19 ;
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 14; :: thesis: verum

end;

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