Theorem vtocle 2823

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 2820	. 2 ⊢ (𝐴 ∈ V → 𝜑)
4	1, 3	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1363   ∈ wcel 2158  Vcvv 2749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751

This theorem is referenced by:  repizf2  4174  nn0ind-raph  9383