Theorem rspcime 2917

Description: Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypotheses

Ref	Expression
rspcime.1	|- ( ( ph /\ x = A ) -> ps )
rspcime.2	|- ( ph -> A e. B )

Assertion

Ref	Expression
rspcime	|- ( ph -> E. x e. B ps )

Distinct variable groups:   ph, x   x, B   x, A

Allowed substitution hint:   ps( x)

Proof of Theorem rspcime

Step	Hyp	Ref	Expression
1		rspcime.2	. 2 |- ( ph -> A e. B )
2		rspcime.1	. . 3 |- ( ( ph /\ x = A ) -> ps )
3		simpl 109	. . 3 |- ( ( ph /\ x = A ) -> ph )
4	2, 3	2thd 175	. 2 |- ( ( ph /\ x = A ) -> ( ps <-> ph ) )
5		id 19	. 2 |- ( ph -> ph )
6	1, 4, 5	rspcedvd 2916	1 |- ( ph -> E. x e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   E.wrex 2511

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804

This theorem is referenced by:  elrnmptdv  4986