Theorem reximdva0

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 )

Proof

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 )