Theorem vtoclegft 3570

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3543.) (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 261	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
2	1	ax-gen 1797	. . . . 5 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
3		ceqsalt 3503	. . . . 5 ⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜑)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
4	2, 3	mp3an2 1449	. . . 4 ⊢ ((Ⅎ𝑥𝜑 ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
5	4	ancoms 459	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
6	5	biimpd 228	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜑))
7	6	3impia 1117	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-clel 2809

This theorem is referenced by:  vtoclefex  36019