Theorem vtoclegft 3529

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3510.) (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 264	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
2	1	ax-gen 1803	. . . 4 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
3		ceqsalt 3466	. . . 4 ⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜑)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
4	2, 3	mp3an2 1458	. . 3 ⊢ ((Ⅎ𝑥𝜑 ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
5	4	ancoms 460	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑))
6	5	biimp3a 1478	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1093  ∀wal 1546   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121

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 1975  ax-7 2016  ax-8 2123  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-an 398  df-3an 1095  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-clel 2816

This theorem is referenced by:  vtoclefex  37711