Theorem rspcime 2885

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 2884	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∃wrex 2486

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775

This theorem is referenced by:  elrnmptdv  4937