let X be BCK-algebra; :: thesis: for I being Ideal of X st ( for x, y, z being Element of X st (x \ y) \ z in I holds
(x \ z) \ (y \ z) in I ) holds
I is positive-implicative-ideal of X
let I be Ideal of X; :: thesis: ( ( for x, y, z being Element of X st (x \ y) \ z in I holds
(x \ z) \ (y \ z) in I ) implies I is positive-implicative-ideal of X )
assume A: for x, y, z being Element of X st (x \ y) \ z in I holds
(x \ z) \ (y \ z) in I ; :: thesis: I is positive-implicative-ideal of X
B: for x, y, z being Element of X st (x \ y) \ z in I & y \ z in I holds
x \ z in I

proof

let x, y, z be Element of X; :: thesis: ( (x \ y) \ z in I & y \ z in I implies x \ z in I )
assume A1: ( (x \ y) \ z in I & y \ z in I ) ; :: thesis: x \ z in I
then (x \ z) \ (y \ z) in I by A;
hence x \ z in I by A1, BCIALG_1:def 18; :: thesis: verum

end;

0. X in I by BCIALG_1:def 18;
hence I is positive-implicative-ideal of X by B, Def5; :: thesis: verum