let Y be non empty set ; :: thesis: for a, b, c, d being Function of Y,BOOLEAN st a 'imp' (b 'imp' c) = I_el Y & a 'imp' (c 'imp' d) = I_el Y holds
a 'imp' (b 'imp' d) = I_el Y
let a, b, c, d be Function of Y,BOOLEAN; :: thesis: ( a 'imp' (b 'imp' c) = I_el Y & a 'imp' (c 'imp' d) = I_el Y implies a 'imp' (b 'imp' d) = I_el Y )
assume that
A1: a 'imp' (b 'imp' c) = I_el Y and
A2: a 'imp' (c 'imp' d) = I_el Y ; :: thesis: a 'imp' (b 'imp' d) = I_el Y
for x being Element of Y holds (a 'imp' (b 'imp' d)) . x = TRUE

proof

let x be Element of Y; :: thesis: (a 'imp' (b 'imp' d)) . x = TRUE
A3: (a 'imp' (b 'imp' d)) . x = ('not' (a . x)) 'or' ((b 'imp' d) . x) by BVFUNC_1:def 8
.= ('not' (a . x)) 'or' (('not' (b . x)) 'or' (d . x)) by BVFUNC_1:def 8 ;
(a 'imp' (c 'imp' d)) . x = TRUE by A2, BVFUNC_1:def 11;
then ('not' (a . x)) 'or' ((c 'imp' d) . x) = TRUE by BVFUNC_1:def 8;
then A4: ('not' (a . x)) 'or' (('not' (c . x)) 'or' (d . x)) = TRUE by BVFUNC_1:def 8;
(a 'imp' (b 'imp' c)) . x = TRUE by A1, BVFUNC_1:def 11;
then ('not' (a . x)) 'or' ((b 'imp' c) . x) = TRUE by BVFUNC_1:def 8;
then A5: ('not' (a . x)) 'or' (('not' (b . x)) 'or' (c . x)) = TRUE by BVFUNC_1:def 8;
A6: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;

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

per cases ( ( 'not' (a . x) = TRUE & 'not' (a . x) = TRUE ) or ( 'not' (a . x) = TRUE & ('not' (c . x)) 'or' (d . x) = TRUE ) or ( ('not' (b . x)) 'or' (c . x) = TRUE & 'not' (a . x) = TRUE ) or ( ('not' (b . x)) 'or' (c . x) = TRUE & ('not' (c . x)) 'or' (d . x) = TRUE ) ) by A5, A6, A4, BINARITH:3;

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

hence (a 'imp' (b 'imp' d)) . x = TRUE by A3, BINARITH:10; :: thesis: verum

end;

case ( 'not' (a . x) = TRUE & ('not' (c . x)) 'or' (d . x) = TRUE ) ; :: thesis: (a 'imp' (b 'imp' d)) . x = TRUE

hence (a 'imp' (b 'imp' d)) . x = TRUE by A3, BINARITH:10; :: thesis: verum

end;

case ( ('not' (b . x)) 'or' (c . x) = TRUE & 'not' (a . x) = TRUE ) ; :: thesis: (a 'imp' (b 'imp' d)) . x = TRUE

hence (a 'imp' (b 'imp' d)) . x = TRUE by A3, BINARITH:10; :: thesis: verum

end;

case A7: ( ('not' (b . x)) 'or' (c . x) = TRUE & ('not' (c . x)) 'or' (d . x) = TRUE ) ; :: thesis: (a 'imp' (b 'imp' d)) . x = TRUE

A8: ( 'not' (c . x) = TRUE or 'not' (c . x) = FALSE ) by XBOOLEAN:def 3;
A9: ( 'not' (b . x) = TRUE or 'not' (b . x) = FALSE ) by XBOOLEAN:def 3;

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

per cases ( ( 'not' (b . x) = TRUE & 'not' (c . x) = TRUE ) or ( 'not' (b . x) = TRUE & d . x = TRUE ) or ( c . x = TRUE & 'not' (c . x) = TRUE ) or ( c . x = TRUE & d . x = TRUE ) ) by A7, A9, A8, BINARITH:3;

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

then (a 'imp' (b 'imp' d)) . x = ('not' (a . x)) 'or' TRUE by A3, BINARITH:10
.= TRUE by BINARITH:10 ;
hence (a 'imp' (b 'imp' d)) . x = TRUE ; :: thesis: verum

end;

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

hence (a 'imp' (b 'imp' d)) . x = TRUE by A3, BINARITH:10; :: thesis: verum

end;

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

hence (a 'imp' (b 'imp' d)) . x = TRUE by MARGREL1:11; :: thesis: verum

end;

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

then (a 'imp' (b 'imp' d)) . x = ('not' (a . x)) 'or' TRUE by A3, BINARITH:10
.= TRUE by BINARITH:10 ;
hence (a 'imp' (b 'imp' d)) . x = TRUE ; :: thesis: verum

end;

end;

end;

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

end;

end;

end;

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

end;

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