Theorem vtocle 3501

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) Avoid df-clab 2718. (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 3449	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
4	1, 3	exlimiiv 1938	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-clel 2814

This theorem is referenced by:  vtocl  3503  vtocl2  3510  vtocl3  3511  zfrepclf  5213  tz6.12i  6853  eloprabga  7465  cfflb  10172  axcc3  10351  nn0ind-raph  12620  finxpreclem6  37758  ormkglobd  47320