Theorem vtoclegft 3543

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3524.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)

Assertion

Ref	Expression
vtoclegft	⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem vtoclegft

Step	Hyp	Ref	Expression
1		biidd 265	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
2	1	ax-gen 1827	. . . 4 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
3		ceqsalt 3483	. . . 4 ⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜑)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
4	2, 3	mp3an2 1478	. . 3 ⊢ ((Ⅎ𝑥𝜑 ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
5	4	ancoms 464	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
6	5	biimp3a 1498	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103  ∀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:  vtoclefex  38258