Description: Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012)
Ref | Expression | ||
---|---|---|---|
Hypothesis | reximdva0.1 | |- ( ( ph /\ x e. A ) -> ps ) |
|
Assertion | reximdva0 | |- ( ( ph /\ A =/= (/) ) -> E. x e. A ps ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reximdva0.1 | |- ( ( ph /\ x e. A ) -> ps ) |
|
2 | n0 | |- ( A =/= (/) <-> E. x x e. A ) |
|
3 | 1 | ex | |- ( ph -> ( x e. A -> ps ) ) |
4 | 3 | ancld | |- ( ph -> ( x e. A -> ( x e. A /\ ps ) ) ) |
5 | 4 | eximdv | |- ( ph -> ( E. x x e. A -> E. x ( x e. A /\ ps ) ) ) |
6 | 5 | imp | |- ( ( ph /\ E. x x e. A ) -> E. x ( x e. A /\ ps ) ) |
7 | 2 6 | sylan2b | |- ( ( ph /\ A =/= (/) ) -> E. x ( x e. A /\ ps ) ) |
8 | df-rex | |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) ) |
|
9 | 7 8 | sylibr | |- ( ( ph /\ A =/= (/) ) -> E. x e. A ps ) |