Theorem reximssdv 2591

Description: Derivation of a restricted existential quantification over a subset (the second hypothesis implies A C_ B), deduction form. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
reximssdv.1	|- ( ph -> E. x e. B ps )
reximssdv.2	|- ( ( ph /\ ( x e. B /\ ps ) ) -> x e. A )
reximssdv.3	|- ( ( ph /\ ( x e. B /\ ps ) ) -> ch )

Assertion

Ref	Expression
reximssdv	|- ( ph -> E. x e. A ch )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)   B( x)

Proof of Theorem reximssdv

Step	Hyp	Ref	Expression
1		reximssdv.1	. 2 |- ( ph -> E. x e. B ps )
2		reximssdv.2	. . . . 5 |- ( ( ph /\ ( x e. B /\ ps ) ) -> x e. A )
3		reximssdv.3	. . . . 5 |- ( ( ph /\ ( x e. B /\ ps ) ) -> ch )
4	2, 3	jca 306	. . . 4 |- ( ( ph /\ ( x e. B /\ ps ) ) -> ( x e. A /\ ch ) )
5	4	ex 115	. . 3 |- ( ph -> ( ( x e. B /\ ps ) -> ( x e. A /\ ch ) ) )
6	5	reximdv2 2586	. 2 |- ( ph -> ( E. x e. B ps -> E. x e. A ch ) )
7	1, 6	mpd 13	1 |- ( ph -> E. x e. A ch )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2158   E.wrex 2466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544

This theorem depends on definitions:  df-bi 117  df-rex 2471

This theorem is referenced by:  suplocexprlemrl  7729  neissex  13936  iscnp4  13989  suplociccex  14374