Theorem axacndlem5 10033

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 10032	. . . 4 ⊢ ∃𝑥∀𝑣∀𝑧(∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑣(∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑣 = 𝑤))
2		nfnae 2456	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
3		nfnae 2456	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥
4		nfnae 2456	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑤
5	2, 3, 4	nf3an 1902	. . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
6		nfnae 2456	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
7		nfnae 2456	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
8		nfnae 2456	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑤
9	6, 7, 8	nf3an 1902	. . . . . 6 ⊢ Ⅎ𝑦(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
10		nfnae 2456	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑧
11		nfnae 2456	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑥
12		nfnae 2456	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑤
13	10, 11, 12	nf3an 1902	. . . . . . 7 ⊢ Ⅎ𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
14		nfcvd 2978	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦𝑣)
15		nfcvf 3007	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
16	15	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦𝑧)
17	14, 16	nfeld 2989	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑣 ∈ 𝑧)
18		nfcvf 3007	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑤 → Ⅎ𝑦𝑤)
19	18	3ad2ant3 1131	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦𝑤)
20	16, 19	nfeld 2989	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑧 ∈ 𝑤)
21	17, 20	nfand 1898	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤))
22	5, 21	nfald 2347	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤))
23		nfnae 2456	. . . . . . . . . 10 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑧
24		nfnae 2456	. . . . . . . . . 10 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑥
25		nfnae 2456	. . . . . . . . . 10 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑤
26	23, 24, 25	nf3an 1902	. . . . . . . . 9 ⊢ Ⅎ𝑤(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
27		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑣(¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤)
28	14, 19	nfeld 2989	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑣 ∈ 𝑤)
29		nfcvf 3007	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
30	29	3ad2ant2 1130	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦𝑥)
31	19, 30	nfeld 2989	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑤 ∈ 𝑥)
32	28, 31	nfand 1898	. . . . . . . . . . . . 13 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
33	21, 32	nfand 1898	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
34	26, 33	nfexd 2348	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦∃𝑤((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
35	14, 19	nfeqd 2988	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑦 𝑣 = 𝑤)
36	34, 35	nfbid 1903	. . . . . . . . . 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 2347	. . . . . . . . 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 2348	. . . . . . . 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 1895	. . . . . . 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 2347	. . . . . 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 2978	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑧𝑣)
42		nfcvf2 3008	. . . . . . . . . . 11 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦)
43	42	3ad2ant1 1129	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑧𝑦)
44	41, 43	nfeqd 2988	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑧 𝑣 = 𝑦)
45	13, 44	nfan1 2200	. . . . . . . 8 ⊢ Ⅎ𝑧((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦)
46		nfcvd 2978	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑥𝑣)
47		nfcvf2 3008	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑥𝑦)
48	47	3ad2ant2 1130	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑥𝑦)
49	46, 48	nfeqd 2988	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑥 𝑣 = 𝑦)
50	5, 49	nfan1 2200	. . . . . . . . . 10 ⊢ Ⅎ𝑥((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦)
51		simpr 487	. . . . . . . . . . . 12 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → 𝑣 = 𝑦)
52	51	eleq1d 2897	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (𝑣 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
53	52	anbi1d 631	. . . . . . . . . 10 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → ((𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤)))
54	50, 53	albid 2224	. . . . . . . . 9 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (∀𝑥(𝑣 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ↔ ∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤)))
55		nfcvd 2978	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑤𝑣)
56		nfcvf2 3008	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∀𝑦 𝑦 = 𝑤 → Ⅎ𝑤𝑦)
57	56	3ad2ant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑤𝑦)
58	55, 57	nfeqd 2988	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) → Ⅎ𝑤 𝑣 = 𝑦)
59	26, 58	nfan1 2200	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦)
60	51	eleq1d 2897	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (𝑣 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
61	60	anbi1d 631	. . . . . . . . . . . . . . . 16 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ↔ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
62	53, 61	anbi12d 632	. . . . . . . . . . . . . . 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 2225	. . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . 14 ⊢ (((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑤) ∧ 𝑣 = 𝑦) → (𝑣 = 𝑤 ↔ 𝑦 = 𝑤))
65	63, 64	bibi12d 348	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 2428	. . . . . . . . . . 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 2225	. . . . . . . . . 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 483	. . . . . . . . 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 347	. . . . . . . 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 2224	. . . . . . 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 415	. . . . . 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 2428	. . . . 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 2225	. . . 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 235	. . 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 1115	. 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 10031	. 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 10029	. . 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 2450	. 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 2455	. . . . 5 ⊢ Ⅎ𝑧∀𝑦 𝑦 = 𝑤
81		en2lp 9069	. . . . . . . . 9 ⊢ ¬ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑦)
82		elequ2 2129	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
83	82	anbi2d 630	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑦) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤)))
84	81, 83	mtbii 328	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ¬ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤))
85	84	sps 2184	. . . . . . 7 ⊢ (∀𝑦 𝑦 = 𝑤 → ¬ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤))
86	85	pm2.21d 121	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑤 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
87	86	spsd 2186	. . . . 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 2213	. . . 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 2341	. . 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	89	19.8ad 2181	. 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)))
91	76, 77, 79, 90	pm2.61iii 187	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 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1535  ∃wex 1780  Ⅎwnfc 2961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-reg 9056  ax-ac 9881

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-eprel 5465  df-fr 5514

This theorem is referenced by:  axacnd  10034