Theorem vtoclf 3511

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2405. (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 235	. 2 ⊢ (𝑥 = 𝐴 → 𝜓)
6	1, 2, 5	vtoclef 3510	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121  Vcvv 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-nf 1792  df-clel 2816

This theorem is referenced by:  summolem2a  15672  prodmolem2a  15894  poimirlem24  38026  poimirlem28  38030  monotuz  43401  oddcomabszz  43404  binomcxplemnotnn0  44815  limclner  46108  climinf2mpt  46171  climinfmpt  46172  dvnmptdivc  46395  dvnmul  46400  salpreimagtge  47182  salpreimaltle  47183