Theorem vtoclgft 2906

Description: Closed theorem form of vtoclgf 2914. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)

Assertion

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

Proof of Theorem vtoclgft

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ V → A ∈ V)
2		elisset 2870	. . . . 5 ⊢ (A ∈ V → ∃z z = A)
3	2	3ad2ant3 978	. . . 4 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → ∃z z = A)
4		nfnfc1 2493	. . . . . . 7 ⊢ ℲxℲxA
5		nfcvd 2491	. . . . . . . 8 ⊢ (ℲxA → Ⅎxz)
6		id 19	. . . . . . . 8 ⊢ (ℲxA → ℲxA)
7	5, 6	nfeqd 2504	. . . . . . 7 ⊢ (ℲxA → Ⅎx z = A)
8		eqeq1 2359	. . . . . . . 8 ⊢ (z = x → (z = A ↔ x = A))
9	8	a1i 10	. . . . . . 7 ⊢ (ℲxA → (z = x → (z = A ↔ x = A)))
10	4, 7, 9	cbvexd 2009	. . . . . 6 ⊢ (ℲxA → (∃z z = A ↔ ∃x x = A))
11	10	ad2antrr 706	. . . . 5 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ)) → (∃z z = A ↔ ∃x x = A))
12	11	3adant3 975	. . . 4 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → (∃z z = A ↔ ∃x x = A))
13	3, 12	mpbid 201	. . 3 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → ∃x x = A)
14		bi1 178	. . . . . . . . 9 ⊢ ((φ ↔ ψ) → (φ → ψ))
15	14	imim2i 13	. . . . . . . 8 ⊢ ((x = A → (φ ↔ ψ)) → (x = A → (φ → ψ)))
16	15	com23 72	. . . . . . 7 ⊢ ((x = A → (φ ↔ ψ)) → (φ → (x = A → ψ)))
17	16	imp 418	. . . . . 6 ⊢ (((x = A → (φ ↔ ψ)) ∧ φ) → (x = A → ψ))
18	17	alanimi 1562	. . . . 5 ⊢ ((∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) → ∀x(x = A → ψ))
19	18	3ad2ant2 977	. . . 4 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → ∀x(x = A → ψ))
20		simp1r 980	. . . . 5 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → Ⅎxψ)
21		19.23t 1800	. . . . 5 ⊢ (Ⅎxψ → (∀x(x = A → ψ) ↔ (∃x x = A → ψ)))
22	20, 21	syl 15	. . . 4 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → (∀x(x = A → ψ) ↔ (∃x x = A → ψ)))
23	19, 22	mpbid 201	. . 3 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → (∃x x = A → ψ))
24	13, 23	mpd 14	. 2 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → ψ)
25	1, 24	syl3an3 1217	1 ⊢ (((ℲxA ∧ Ⅎxψ) ∧ (∀x(x = A → (φ ↔ ψ)) ∧ ∀xφ) ∧ A ∈ V) → ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  Vcvv 2860

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-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  vtocldf  2907  iota2df  4366