Theorem vtoclef 3510

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 3451	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtoclef.1	. . 3 ⊢ Ⅎ𝑥𝜑
4		vtoclef.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
5	3, 4	exlimi 2231	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
6	2, 5	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1548  ∃wex 1787  Ⅎwnf 1791   ∈ wcel 2121  Vcvv 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-nf 1792  df-clel 2816

This theorem is referenced by:  vtoclf  3511  nn0ind-raph  12624  rdgssun  37755  finxpreclem2  37767