Theorem vtoclegft 2927

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2928.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

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		elisset 2870	. . . 4 ⊢ (A ∈ B → ∃x x = A)
2		exim 1575	. . . 4 ⊢ (∀x(x = A → φ) → (∃x x = A → ∃xφ))
3	1, 2	mpan9 455	. . 3 ⊢ ((A ∈ B ∧ ∀x(x = A → φ)) → ∃xφ)
4	3	3adant2 974	. 2 ⊢ ((A ∈ B ∧ Ⅎxφ ∧ ∀x(x = A → φ)) → ∃xφ)
5		19.9t 1779	. . 3 ⊢ (Ⅎxφ → (∃xφ ↔ φ))
6	5	3ad2ant2 977	. 2 ⊢ ((A ∈ B ∧ Ⅎxφ ∧ ∀x(x = A → φ)) → (∃xφ ↔ φ))
7	4, 6	mpbid 201	1 ⊢ ((A ∈ B ∧ Ⅎxφ ∧ ∀x(x = A → φ)) → φ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by: (None)