Theorem rspcime 2841

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 108	. . 3 |- ( ( ph /\ x = A ) -> ph )
4	2, 3	2thd 174	. 2 |- ( ( ph /\ x = A ) -> ( ps <-> ph ) )
5		id 19	. 2 |- ( ph -> ph )
6	1, 4, 5	rspcedvd 2840	1 |- ( ph -> E. x e. B ps )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732

This theorem is referenced by:  elrnmptdv  4865