Theorem rspcime 2841

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 108	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜑)
4	2, 3	2thd 174	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜑))
5		id 19	. 2 ⊢ (𝜑 → 𝜑)
6	1, 4, 5	rspcedvd 2840	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ∃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