Theorem rspcime 3595

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

Hypotheses

Ref	Expression
rspcime.1	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)
rspcime.2	⊢ (𝜑 → 𝐴 ∈ 𝐵)

Assertion

Ref	Expression
rspcime	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

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

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcime

Step	Hyp	Ref	Expression
1		rspcime.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		rspcime.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)
3		simpl 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜑)
4	2, 3	2thd 268	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜑))
5		id 23	. 2 ⊢ (𝜑 → 𝜑)
6	1, 4, 5	rspcedvd 3592	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096

This theorem is referenced by:  rspcedeq1vd  3597  rspcedeq2vd  3598  elrnmptdv  5956  aks4d1p8d2  42776  mnuprdlem3  44910  mnurndlem1  44917  grumnudlem  44921  grumnud  44922  inaex  44933  gruex  44934