Theorem vtoclf 3539

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2433. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)

Hypotheses

Ref	Expression
vtoclf.1	⊢ Ⅎ𝑥𝜓
vtoclf.2	⊢ 𝐴 ∈ V
vtoclf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtoclf.4	⊢ 𝜑

Assertion

Ref	Expression
vtoclf	⊢ 𝜓

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem vtoclf

Step	Hyp	Ref	Expression
1		vtoclf.1	. 2 ⊢ Ⅎ𝑥𝜓
2		vtoclf.2	. 2 ⊢ 𝐴 ∈ V
3		vtoclf.4	. . 3 ⊢ 𝜑
4		vtoclf.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	mpbii 236	. 2 ⊢ (𝑥 = 𝐴 → 𝜓)
6	1, 2, 5	vtoclef 3538	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  Ⅎwnf 1810   ∈ wcel 2149  Vcvv 3463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811  df-clel 2844

This theorem is referenced by:  summolem2a  15766  prodmolem2a  15988  poimirlem24  38217  poimirlem28  38221  monotuz  43594  oddcomabszz  43597  binomcxplemnotnn0  44992  limclner  46291  climinf2mpt  46354  climinfmpt  46355  dvnmptdivc  46578  dvnmul  46583  salpreimagtge  47365  salpreimaltle  47366