Theorem axacndlem5 9433

Description: Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Assertion

Ref	Expression
axacndlem5	⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))

Distinct variable group:   𝑧,𝑤

Proof of Theorem axacndlem5

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axacndlem4 9432	. . . 4 ⊢ ∃𝑥∀𝑣∀𝑧(∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤))
2		nfnae 2318	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
3		nfnae 2318	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥
4		nfnae 2318	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑤
5	2, 3, 4	nf3an 1831	. . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
6		nfnae 2318	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
7		nfnae 2318	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
8		nfnae 2318	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑤
9	6, 7, 8	nf3an 1831	. . . . . 6 ⊢ Ⅎ𝑦(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
10		nfnae 2318	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑧
11		nfnae 2318	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑥
12		nfnae 2318	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑤
13	10, 11, 12	nf3an 1831	. . . . . . 7 ⊢ Ⅎ𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
14		nfcvd 2765	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦𝑣)
15		nfcvf 2788	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
16	15	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦𝑧)
17	14, 16	nfeld 2773	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑣 ∈ 𝑧)
18		nfcvf 2788	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑤 → Ⅎ𝑦𝑤)
19	18	3ad2ant3 1084	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦𝑤)
20	16, 19	nfeld 2773	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑧 ∈ 𝑤)
21	17, 20	nfand 1826	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤))
22	5, 21	nfald 2165	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤))
23		nfnae 2318	. . . . . . . . . 10 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑧
24		nfnae 2318	. . . . . . . . . 10 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑥
25		nfnae 2318	. . . . . . . . . 10 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑤
26	23, 24, 25	nf3an 1831	. . . . . . . . 9 ⊢ Ⅎ𝑤(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
27		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑣(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
28	14, 19	nfeld 2773	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑣 ∈ 𝑤)
29		nfcvf 2788	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
30	29	3ad2ant2 1083	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦𝑥)
31	19, 30	nfeld 2773	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑤 ∈ 𝑥)
32	28, 31	nfand 1826	. . . . . . . . . . . . 13 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
33	21, 32	nfand 1826	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
34	26, 33	nfexd 2167	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
35	14, 19	nfeqd 2772	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑣 = 𝑤)
36	34, 35	nfbid 1832	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤))
37	27, 36	nfald 2165	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤))
38	26, 37	nfexd 2167	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤))
39	22, 38	nfimd 1823	. . . . . . 7 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦(∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤)))
40	13, 39	nfald 2165	. . . . . 6 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦∀𝑧(∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤)))
41		nfcvd 2765	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑧𝑣)
42		nfcvf2 2789	. . . . . . . . . . 11 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦)
43	42	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑧𝑦)
44	41, 43	nfeqd 2772	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑧 𝑣 = 𝑦)
45	13, 44	nfan1 2068	. . . . . . . 8 ⊢ Ⅎ𝑧((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦)
46		nfcvd 2765	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑥𝑣)
47		nfcvf2 2789	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑥𝑦)
48	47	3ad2ant2 1083	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑥𝑦)
49	46, 48	nfeqd 2772	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑥 𝑣 = 𝑦)
50	5, 49	nfan1 2068	. . . . . . . . . 10 ⊢ Ⅎ𝑥((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦)
51		simpr 477	. . . . . . . . . . . 12 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → 𝑣 = 𝑦)
52	51	eleq1d 2686	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (𝑣 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
53	52	anbi1d 741	. . . . . . . . . 10 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → ((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤)))
54	50, 53	albid 2090	. . . . . . . . 9 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ↔ ∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤)))
55		nfcvd 2765	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑤𝑣)
56		nfcvf2 2789	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∀𝑦 𝑦 = 𝑤 → Ⅎ𝑤𝑦)
57	56	3ad2ant3 1084	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑤𝑦)
58	55, 57	nfeqd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑤 𝑣 = 𝑦)
59	26, 58	nfan1 2068	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦)
60	51	eleq1d 2686	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (𝑣 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
61	60	anbi1d 741	. . . . . . . . . . . . . . . 16 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ↔ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
62	53, 61	anbi12d 747	. . . . . . . . . . . . . . 15 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))))
63	59, 62	exbid 2091	. . . . . . . . . . . . . 14 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ ∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))))
64	51	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (𝑣 = 𝑤 ↔ 𝑦 = 𝑤))
65	63, 64	bibi12d 335	. . . . . . . . . . . . 13 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → ((∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤) ↔ (∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
66	65	ex 450	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → (𝑣 = 𝑦 → ((∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤) ↔ (∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))))
67	9, 36, 66	cbvald 2277	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → (∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤) ↔ ∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
68	26, 67	exbid 2091	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → (∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤) ↔ ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
69	68	adantr 481	. . . . . . . . 9 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤) ↔ ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
70	54, 69	imbi12d 334	. . . . . . . 8 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → ((∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤)) ↔ (∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))))
71	45, 70	albid 2090	. . . . . . 7 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (∀𝑧(∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤)) ↔ ∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))))
72	71	ex 450	. . . . . 6 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → (𝑣 = 𝑦 → (∀𝑧(∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤)) ↔ ∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))))
73	9, 40, 72	cbvald 2277	. . . . 5 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → (∀𝑣∀𝑧(∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤)) ↔ ∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))))
74	5, 73	exbid 2091	. . . 4 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → (∃𝑥∀𝑣∀𝑧(∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤)) ↔ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))))
75	1, 74	mpbii 223	. . 3 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
76	75	3exp 1264	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑤 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))))
77		axacndlem3 9431	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
78		axacndlem1 9429	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
79	78	aecoms 2312	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
80		nfae 2316	. . . . 5 ⊢ Ⅎ𝑧∀𝑦 𝑦 = 𝑤
81		en2lp 8510	. . . . . . . . 9 ⊢ ¬ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑦)
82		elequ2 2004	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
83	82	anbi2d 740	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑦) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤)))
84	81, 83	mtbii 316	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ¬ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤))
85	84	sps 2055	. . . . . . 7 ⊢ (∀𝑦 𝑦 = 𝑤 → ¬ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤))
86	85	pm2.21d 118	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑤 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
87	86	spsd 2057	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑤 → (∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
88	80, 87	alrimi 2082	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑤 → ∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
89	88	axc4i 2131	. . 3 ⊢ (∀𝑦 𝑦 = 𝑤 → ∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
90		19.8a 2052	. . 3 ⊢ (∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)) → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
91	89, 90	syl 17	. 2 ⊢ (∀𝑦 𝑦 = 𝑤 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
92	76, 77, 79, 91	pm2.61iii 179	1 ⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481  ∃wex 1704  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497  ax-ac 9281

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073

This theorem is referenced by:  axacnd  9434