Theorem vtocle 3555

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) Avoid df-clab 2715. (Revised by Wolf Lammen, 31-May-2025.)

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.2	. 2 ⊢ (𝑥 = 𝐴 → 𝜑)
2		vtocle.1	. . 3 ⊢ 𝐴 ∈ V
3	2	isseti 3498	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
4	1, 3	exlimiiv 1931	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480

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 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-clel 2816

This theorem is referenced by:  vtocl  3558  zfrepclf  5291  tz6.12i  6934  eloprabga  7542  cfflb  10299  axcc3  10478  nn0ind-raph  12718  finxpreclem6  37397  ormkglobd  46890  tworepnotupword  46901