Description: Restricted quantifier version of one direction of 19.28 . (The other direction holds when A is nonempty, see r19.28zv .) (Contributed by NM, 2-Apr-2004) (Proof shortened by Wolf Lammen, 17-Jun-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | r19.28v | |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id | |- ( ph -> ph ) |
|
2 | 1 | ralrimivw | |- ( ph -> A. x e. A ph ) |
3 | 2 | anim1i | |- ( ( ph /\ A. x e. A ps ) -> ( A. x e. A ph /\ A. x e. A ps ) ) |
4 | r19.26 | |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) |
|
5 | 3 4 | sylibr | |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) ) |