Theorem rr-spce 41815

Description: Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)

Hypotheses

Ref	Expression
rr-spce.1	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)
rr-spce.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
rr-spce	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem rr-spce

Step	Hyp	Ref	Expression
1		rr-spce.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	elexd 3452	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
3		isset 3445	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4	2, 3	sylib 217	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
5		rr-spce.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)
6	5	ex 413	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝜓))
7	6	eximdv 1920	. 2 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜓))
8	4, 7	mpd 15	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434

This theorem is referenced by:  grumnudlem  41903