Theorem rspcime 2875

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 2874	1 |- ( ph -> E. x e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   E.wrex 2476

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765

This theorem is referenced by:  elrnmptdv  4920