Theorem rspcime 2848

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 2847	1 |- ( ph -> E. x e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739

This theorem is referenced by:  elrnmptdv  4880