Theorem vtocle 3522

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) Avoid df-clab 2742. (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 3472	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
4	1, 3	exlimiiv 1966	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2839

This theorem is used by:  vtocl  3524  vtocl2  3530  vtocl3  3531  zfrepclf  5250  tz6.12i  6942  eloprabga  7560  cfflb  10376  axcc3  10555  nn0ind-raph  12839  finxpreclem6  38587  ormkglobd  48160