Theorem rspcime 2927

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 109	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜑)
4	2, 3	2thd 175	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜑))
5		id 19	. 2 ⊢ (𝜑 → 𝜑)
6	1, 4, 5	rspcedvd 2926	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∃wrex 2521

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814

This theorem is referenced by:  elrnmptdv  5010