Theorem vtoclefex 38258

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 1827	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝜑)
4		vtoclegft 3543	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)
5	1, 3, 4	mp3an23 1482	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-clel 2835

This theorem is used by:  finxpreclem2  38314