Theorem vtocle 3518

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) Avoid df-clab 2739. (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 3468	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
4	1, 3	exlimiiv 1964	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450

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 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2835

This theorem is used by:  vtocl  3520  vtocl2  3526  vtocl3  3527  zfrepclf  5246  tz6.12i  6905  eloprabga  7523  cfflb  10262  axcc3  10441  nn0ind-raph  12722  finxpreclem6  38151  ormkglobd  47706