Theorem vtoclegft 2875

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2876.) (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 2814	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
2		exim 1645	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑))
3	1, 2	mpan9 281	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → ∃𝑥𝜑)
4	3	3adant2 1040	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → ∃𝑥𝜑)
5		19.9t 1688	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
6	5	3ad2ant2 1043	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → (∃𝑥𝜑 ↔ 𝜑))
7	4, 6	mpbid 147	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1002  ∀wal 1393   = wceq 1395  Ⅎwnf 1506  ∃wex 1538   ∈ wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by: (None)