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

proof

let x be Element of Y; :: thesis: (a 'imp' c) . x = TRUE
(a 'imp' b) . x = TRUE by A1, BVFUNC_1:def 14;
then A2: ('not' (a . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 11;
A3: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;
(b 'imp' c) . x = TRUE by A1, BVFUNC_1:def 14;
then A4: ('not' (b . x)) 'or' (c . x) = TRUE by BVFUNC_1:def 11;
A5: ( 'not' (b . x) = TRUE or 'not' (b . x) = FALSE ) by XBOOLEAN:def 3;
A6: (a 'imp' c) . x = ('not' (a . x)) 'or' (c . x) by BVFUNC_1:def 11;

now

per cases ( ( 'not' (a . x) = TRUE & 'not' (b . x) = TRUE ) or ( 'not' (a . x) = TRUE & c . x = TRUE ) or ( b . x = TRUE & 'not' (b . x) = TRUE ) or ( b . x = TRUE & c . x = TRUE ) ) by A2, A3, A4, A5, BINARITH:7;

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

hence (a 'imp' c) . x = TRUE by A6, BINARITH:19; :: thesis: verum

end;

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

hence (a 'imp' c) . x = TRUE by A6; :: thesis: verum

end;

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

hence (a 'imp' c) . x = TRUE by MARGREL1:41; :: thesis: verum

end;

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

hence (a 'imp' c) . x = TRUE by A6, BINARITH:19; :: thesis: verum

end;

end;

end;

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

end;

hence a 'imp' c = I_el Y by BVFUNC_1:def 14; :: thesis: verum