Theorem vtoclgft 2814

Description: Closed theorem form of vtoclgf 2822. (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 2774	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elisset 2777	. . . . 5 ⊢ (𝐴 ∈ V → ∃𝑧 𝑧 = 𝐴)
3	2	3ad2ant3 1022	. . . 4 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∃𝑧 𝑧 = 𝐴)
4		nfnfc1 2342	. . . . . . 7 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
5		nfcvd 2340	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑧)
6		id 19	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
7	5, 6	nfeqd 2354	. . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 = 𝐴)
8		eqeq1 2203	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝐴 ↔ 𝑥 = 𝐴))
9	8	a1i 9	. . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → (𝑧 = 𝑥 → (𝑧 = 𝐴 ↔ 𝑥 = 𝐴)))
10	4, 7, 9	cbvexd 1942	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
11	10	ad2antrr 488	. . . . 5 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑)) → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
12	11	3adant3 1019	. . . 4 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
13	3, 12	mpbid 147	. . 3 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∃𝑥 𝑥 = 𝐴)
14		biimp 118	. . . . . . . . 9 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
15	14	imim2i 12	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜑 → 𝜓)))
16	15	com23 78	. . . . . . 7 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜑 → (𝑥 = 𝐴 → 𝜓)))
17	16	imp 124	. . . . . 6 ⊢ (((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝜑) → (𝑥 = 𝐴 → 𝜓))
18	17	alanimi 1473	. . . . 5 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓))
19	18	3ad2ant2 1021	. . . 4 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∀𝑥(𝑥 = 𝐴 → 𝜓))
20		simp1r 1024	. . . . 5 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → Ⅎ𝑥𝜓)
21		19.23t 1691	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
22	20, 21	syl 14	. . . 4 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
23	19, 22	mpbid 147	. . 3 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → (∃𝑥 𝑥 = 𝐴 → 𝜓))
24	13, 23	mpd 13	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → 𝜓)
25	1, 24	syl3an3 1284	1 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980  ∀wal 1362   = wceq 1364  Ⅎwnf 1474  ∃wex 1506   ∈ wcel 2167  Ⅎwnfc 2326  Vcvv 2763

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  vtocldf  2815