Theorem vtoclef 3531

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 3455	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtoclef.1	. . 3 ⊢ Ⅎ𝑥𝜑
4		vtoclef.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
5	3, 4	exlimi 2215	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
6	2, 5	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1538  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2111  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870

This theorem is referenced by:  nn0ind-raph  12070  rdgssun  34795  finxpreclem2  34807