Theorem vtocle 3587

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 3584	. 2 ⊢ (𝐴 ∈ V → 𝜑)
4	1, 3	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113  Vcvv 3497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-cleq 2817  df-clel 2896

This theorem is referenced by:  zfrepclf  5201  tz6.12i  6699  eloprabga  7264  cfflb  9684  axcc3  9863  nn0ind-raph  12085  finxpreclem6  34681