Theorem reximdva0m 3430

Description: Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)

Hypothesis

Ref	Expression
reximdva0m.1	|- ( ( ph /\ x e. A ) -> ps )

Assertion

Ref	Expression
reximdva0m	|- ( ( ph /\ E. x x e. A ) -> E. x e. A ps )

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem reximdva0m

Step	Hyp	Ref	Expression
1		reximdva0m.1	. . . . . 6 |- ( ( ph /\ x e. A ) -> ps )
2	1	ex 114	. . . . 5 |- ( ph -> ( x e. A -> ps ) )
3	2	ancld 323	. . . 4 |- ( ph -> ( x e. A -> ( x e. A /\ ps ) ) )
4	3	eximdv 1873	. . 3 |- ( ph -> ( E. x x e. A -> E. x ( x e. A /\ ps ) ) )
5	4	imp 123	. 2 |- ( ( ph /\ E. x x e. A ) -> E. x ( x e. A /\ ps ) )
6		df-rex 2454	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
7	5, 6	sylibr 133	1 |- ( ( ph /\ E. x x e. A ) -> E. x e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1485   e. wcel 2141   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-rex 2454

This theorem is referenced by: (None)