let Y be non empty set ; :: thesis: for a, b, c, d being Element of Funcs 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 Element of Funcs 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 A1: ( a 'imp' (b 'imp' c) = I_el Y & 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
(a 'imp' (b 'imp' c)) . x = TRUE by A1, BVFUNC_1:def 14;
then ('not' (a . x)) 'or' ((b 'imp' c) . x) = TRUE by BVFUNC_1:def 11;
then A2: ('not' (a . x)) 'or' (('not' (b . x)) 'or' (c . x)) = TRUE by BVFUNC_1:def 11;
A3: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;
(a 'imp' (c 'imp' d)) . x = TRUE by A1, BVFUNC_1:def 14;
then ('not' (a . x)) 'or' ((c 'imp' d) . x) = TRUE by BVFUNC_1:def 11;
then A4: ('not' (a . x)) 'or' (('not' (c . x)) 'or' (d . x)) = TRUE by BVFUNC_1:def 11;
A5: (a 'imp' (b 'imp' d)) . x = ('not' (a . x)) 'or' ((b 'imp' d) . x) by BVFUNC_1:def 11
.= ('not' (a . x)) 'or' (('not' (b . x)) 'or' (d . x)) by BVFUNC_1:def 11 ;

now

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 A2, A3, A4, BINARITH:7;

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 A5, BINARITH:19; :: 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 A5, BINARITH:19; :: 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 A5, BINARITH:19; :: thesis: verum

end;

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

A7: ( 'not' (b . x) = TRUE or 'not' (b . x) = FALSE ) by XBOOLEAN:def 3;
A8: ( 'not' (c . x) = TRUE or 'not' (c . x) = FALSE ) by XBOOLEAN:def 3;

now

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 A6, A7, A8, BINARITH:7;

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 A5, BINARITH:19
.= TRUE by BINARITH:19 ;
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 A5, BINARITH:19; :: 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 MARGREL1:41; :: 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 A5, BINARITH:19
.= TRUE by BINARITH:19 ;
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 14; :: thesis: verum