Theorem rr-spce 44988

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 3480	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
3		isset 3471	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4	2, 3	sylib 221	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
5		rr-spce.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)
6	5	ex 418	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝜓))
7	6	eximdv 1950	. 2 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜓))
8	4, 7	mpd 16	1 ⊢ (𝜑 → ∃𝑥𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is used by:  grumnudlem  45055