Theorem vtoclef 3254

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)

Hypotheses

Ref	Expression
vtoclef.1	⊢ Ⅎ𝑥𝜑
vtoclef.2	⊢ 𝐴 ∈ V
vtoclef.3	⊢ (𝑥 = 𝐴 → 𝜑)

Assertion

Ref	Expression
vtoclef	⊢ 𝜑

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclef

Step	Hyp	Ref	Expression
1		vtoclef.2	. . 3 ⊢ 𝐴 ∈ V
2	1	isseti 3182	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtoclef.1	. . 3 ⊢ Ⅎ𝑥𝜑
4		vtoclef.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
5	3, 4	exlimi 2073	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
6	2, 5	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1475  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175

This theorem is referenced by:  nn0ind-raph  11353  finxpreclem2  32403