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

proof

let x be Element of Y; :: thesis: (c 'imp' (a '&' b)) . x = TRUE
(c 'imp' a) . x = TRUE by A1, BVFUNC_1:def 11;
then A3: ('not' (c . x)) 'or' (a . x) = TRUE by BVFUNC_1:def 8;
(c 'imp' b) . x = TRUE by A2, BVFUNC_1:def 11;
then A4: ('not' (c . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 8;
(c 'imp' (a '&' b)) . x = ('not' (c . x)) 'or' ((a '&' b) . x) by BVFUNC_1:def 8
.= ('not' (c . x)) 'or' ((a . x) '&' (b . x)) by MARGREL1:def 20
.= TRUE '&' TRUE by A3, A4, XBOOLEAN:9
.= TRUE ;
hence (c 'imp' (a '&' b)) . x = TRUE ; :: thesis: verum

end;

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