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

proof

let x be Element of Y; :: thesis: (c 'imp' a) . x = TRUE
(c 'imp' (b 'imp' a)) . x = TRUE by A1, BVFUNC_1:def 11;
then ('not' (c . x)) 'or' ((b 'imp' a) . x) = TRUE by BVFUNC_1:def 8;
then A3: ('not' (c . x)) 'or' (('not' (b . x)) 'or' (a . x)) = TRUE by BVFUNC_1:def 8;
('not' (c . x)) 'or' (FALSE 'or' (a . x)) = TRUE by A3, A2, BVFUNC_1:def 11, MARGREL1:11;
then ('not' (c . x)) 'or' (a . x) = TRUE by BINARITH:3;
hence (c 'imp' a) . x = TRUE by BVFUNC_1:def 8; :: thesis: verum

end;

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