Theorem vtocle 3524

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)

Hypotheses

Ref	Expression
vtocle.1	⊢ 𝐴 ∈ V
vtocle.2	⊢ (𝑥 = 𝐴 → 𝜑)

Assertion

Ref	Expression
vtocle	⊢ 𝜑

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem vtocle

Step	Hyp	Ref	Expression
1		vtocle.1	. 2 ⊢ 𝐴 ∈ V
2		vtocle.2	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	vtocleg 3521	. 2 ⊢ (𝐴 ∈ V → 𝜑)
4	1, 3	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1539   ∈ 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

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

This theorem is referenced by:  zfrepclf  5218  tz6.12i  6800  eloprabga  7382  eloprabgaOLD  7383  cfflb  10015  axcc3  10194  nn0ind-raph  12420  finxpreclem6  35567  tworepnotupword  46521