let X be BCI-algebra; :: thesis: for I being Ideal of X holds
( I is p-ideal of X iff for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds
x \ y in I )
let I be Ideal of X; :: thesis: ( I is p-ideal of X iff for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds
x \ y in I )
thus ( I is p-ideal of X implies for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds
x \ y in I ) :: thesis: ( ( for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds
x \ y in I ) implies I is p-ideal of X )

proof

assume A: I is p-ideal of X ; :: thesis: for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds
x \ y in I
let x, y, z be Element of X; :: thesis: ( (x \ z) \ (y \ z) in I implies x \ y in I )
assume A2: (x \ z) \ (y \ z) in I ; :: thesis: x \ y in I
((x \ z) \ (y \ z)) \ (x \ y) = 0. X by BCIALG_1:def 3;
then (x \ z) \ (y \ z) <= x \ y by BCIALG_1:def 11;
hence x \ y in I by A2, TL358, A; :: thesis: verum

end;

assume B: for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds
x \ y in I ; :: thesis: I is p-ideal of X
B1: 0. X in I by BCIALG_1:def 18;
for x, y, z being Element of X st (x \ z) \ (y \ z) in I & y in I holds
x in I

proof

let x, y, z be Element of X; :: thesis: ( (x \ z) \ (y \ z) in I & y in I implies x in I )
assume C1: ( (x \ z) \ (y \ z) in I & y in I ) ; :: thesis: x in I
x \ y in I by B, C1;
hence x in I by C1, BCIALG_1:def 18; :: thesis: verum

end;

hence I is p-ideal of X by B1, Def1; :: thesis: verum