Theorem vtoclgft 3539

Description: Closed theorem form of vtoclgf 3556. The reverse implication is proven in ceqsal1t 3503. See ceqsalt 3504 for a version with 𝑥 and 𝐴 disjoint. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) Avoid ax-13 2369. (Revised by Gino Giotto, 6-Oct-2023.)

Assertion

Ref	Expression
vtoclgft	⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓)

Proof of Theorem vtoclgft

Step	Hyp	Ref	Expression
1		elex 3491	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		issetft 3486	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
3	1, 2	imbitrid 243	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴))
4	3	ad2antrr 722	. . 3 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑)) → (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴))
5	4	3impia 1115	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
6		biimp 214	. . . . . . . . 9 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
7	6	imim2i 16	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜑 → 𝜓)))
8	7	com23 86	. . . . . . 7 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜑 → (𝑥 = 𝐴 → 𝜓)))
9	8	imp 405	. . . . . 6 ⊢ (((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝜑) → (𝑥 = 𝐴 → 𝜓))
10	9	alanimi 1816	. . . . 5 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓))
11		19.23t 2201	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
12	11	adantl 480	. . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
13	10, 12	imbitrid 243	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) → ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
14	13	imp 405	. . 3 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑)) → (∃𝑥 𝑥 = 𝐴 → 𝜓))
15	14	3adant3 1130	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → (∃𝑥 𝑥 = 𝐴 → 𝜓))
16	5, 15	mpd 15	1 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085  ∀wal 1537   = wceq 1539  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2104  Ⅎwnfc 2881  Vcvv 3472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-v 3474

This theorem is referenced by:  vtocldf  3545  bj-vtoclgfALT  36243