Theorem vtocle 3523

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 3473	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
4	1, 3	exlimiiv 1961	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-clel 2838

This theorem is referenced by:  vtocl  3525  vtocl2  3531  vtocl3  3532  zfrepclf  5252  tz6.12i  6907  eloprabga  7519  cfflb  10238  axcc3  10417  nn0ind-raph  12691  finxpreclem6  38042  ormkglobd  47591