Theorem vtoclefex 37277

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 1790	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝜑)
4		vtoclegft 3588	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)
5	1, 3, 4	mp3an23 1451	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1533   = wceq 1535  Ⅎwnf 1778   ∈ wcel 2104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-12 2173

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1087  df-tru 1538  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2711  df-clel 2812

This theorem is referenced by:  finxpreclem2  37333