Theorem rspcime 3585

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 488	. . 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 3582	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091

This theorem is used by:  rspcedeqvd  3587  elrnmptdv  5953  aks4d1p8d2  43326  mnuprdlem3  45448  mnurndlem1  45455  grumnudlem  45459  grumnud  45460  inaex  45471  gruex  45472  chnsubseqword  48091