Theorem vtoclefex 35411

Description: Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)

Hypotheses

Ref	Expression
vtoclefex.1	⊢ Ⅎ𝑥𝜑
vtoclefex.3	⊢ (𝑥 = 𝐴 → 𝜑)

Assertion

Ref	Expression
vtoclefex	⊢ (𝐴 ∈ 𝑉 → 𝜑)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem vtoclefex

Step	Hyp	Ref	Expression
1		vtoclefex.1	. 2 ⊢ Ⅎ𝑥𝜑
2		vtoclefex.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	ax-gen 1803	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝜑)
4		vtoclegft 3513	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)
5	1, 3, 4	mp3an23 1455	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1541   = wceq 1543  Ⅎwnf 1791   ∈ wcel 2112

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 1976  ax-7 2016  ax-8 2114  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-clel 2818

This theorem is referenced by:  finxpreclem2  35467