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Theorem axacnd 9646
Description: A version of 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
axacnd 𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Proof of Theorem axacnd
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 axacndlem5 9645 . . . 4 𝑥𝑦𝑣(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
2 nfnae 2460 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑥
3 nfnae 2460 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑦
4 nfnae 2460 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑤
52, 3, 4nf3an 1980 . . . . 5 𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤)
6 nfnae 2460 . . . . . . 7 𝑦 ¬ ∀𝑧 𝑧 = 𝑥
7 nfnae 2460 . . . . . . 7 𝑦 ¬ ∀𝑧 𝑧 = 𝑦
8 nfnae 2460 . . . . . . 7 𝑦 ¬ ∀𝑧 𝑧 = 𝑤
96, 7, 8nf3an 1980 . . . . . 6 𝑦(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤)
10 nfnae 2460 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
11 nfnae 2460 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
12 nfnae 2460 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑤
1310, 11, 12nf3an 1980 . . . . . . 7 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤)
14 nfcvf 2926 . . . . . . . . . . . 12 (¬ ∀𝑧 𝑧 = 𝑦𝑧𝑦)
15143ad2ant2 1129 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑧𝑦)
16 nfcvd 2903 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑧𝑣)
1715, 16nfeld 2911 . . . . . . . . . 10 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑦𝑣)
18 nfcvf 2926 . . . . . . . . . . . 12 (¬ ∀𝑧 𝑧 = 𝑤𝑧𝑤)
19183ad2ant3 1130 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑧𝑤)
2016, 19nfeld 2911 . . . . . . . . . 10 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑣𝑤)
2117, 20nfand 1975 . . . . . . . . 9 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧(𝑦𝑣𝑣𝑤))
225, 21nfald 2310 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧𝑥(𝑦𝑣𝑣𝑤))
23 nfnae 2460 . . . . . . . . . 10 𝑤 ¬ ∀𝑧 𝑧 = 𝑥
24 nfnae 2460 . . . . . . . . . 10 𝑤 ¬ ∀𝑧 𝑧 = 𝑦
25 nfnae 2460 . . . . . . . . . 10 𝑤 ¬ ∀𝑧 𝑧 = 𝑤
2623, 24, 25nf3an 1980 . . . . . . . . 9 𝑤(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤)
2715, 19nfeld 2911 . . . . . . . . . . . . . 14 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑦𝑤)
28 nfcvf 2926 . . . . . . . . . . . . . . . 16 (¬ ∀𝑧 𝑧 = 𝑥𝑧𝑥)
29283ad2ant1 1128 . . . . . . . . . . . . . . 15 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑧𝑥)
3019, 29nfeld 2911 . . . . . . . . . . . . . 14 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑤𝑥)
3127, 30nfand 1975 . . . . . . . . . . . . 13 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧(𝑦𝑤𝑤𝑥))
3221, 31nfand 1975 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)))
3326, 32nfexd 2312 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)))
3415, 19nfeqd 2910 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑦 = 𝑤)
3533, 34nfbid 1981 . . . . . . . . . 10 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
369, 35nfald 2310 . . . . . . . . 9 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
3726, 36nfexd 2312 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
3822, 37nfimd 1972 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
39 nfcvd 2903 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑥𝑣)
40 nfcvf2 2927 . . . . . . . . . . . . 13 (¬ ∀𝑧 𝑧 = 𝑥𝑥𝑧)
41403ad2ant1 1128 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑥𝑧)
4239, 41nfeqd 2910 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑥 𝑣 = 𝑧)
435, 42nfan1 2215 . . . . . . . . . 10 𝑥((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧)
44 simpr 479 . . . . . . . . . . . 12 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧)
4544eleq2d 2825 . . . . . . . . . . 11 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (𝑦𝑣𝑦𝑧))
4644eleq1d 2824 . . . . . . . . . . 11 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (𝑣𝑤𝑧𝑤))
4745, 46anbi12d 749 . . . . . . . . . 10 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → ((𝑦𝑣𝑣𝑤) ↔ (𝑦𝑧𝑧𝑤)))
4843, 47albid 2237 . . . . . . . . 9 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (∀𝑥(𝑦𝑣𝑣𝑤) ↔ ∀𝑥(𝑦𝑧𝑧𝑤)))
49 nfcvd 2903 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑤𝑣)
50 nfcvf2 2927 . . . . . . . . . . . . 13 (¬ ∀𝑧 𝑧 = 𝑤𝑤𝑧)
51503ad2ant3 1130 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑤𝑧)
5249, 51nfeqd 2910 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑤 𝑣 = 𝑧)
5326, 52nfan1 2215 . . . . . . . . . 10 𝑤((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧)
54 nfcvd 2903 . . . . . . . . . . . . 13 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑦𝑣)
55 nfcvf2 2927 . . . . . . . . . . . . . 14 (¬ ∀𝑧 𝑧 = 𝑦𝑦𝑧)
56553ad2ant2 1129 . . . . . . . . . . . . 13 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑦𝑧)
5754, 56nfeqd 2910 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑦 𝑣 = 𝑧)
589, 57nfan1 2215 . . . . . . . . . . 11 𝑦((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧)
5947anbi1d 743 . . . . . . . . . . . . 13 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥))))
6053, 59exbid 2238 . . . . . . . . . . . 12 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥))))
6160bibi1d 332 . . . . . . . . . . 11 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → ((∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ (∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
6258, 61albid 2237 . . . . . . . . . 10 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (∀𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ∀𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
6353, 62exbid 2238 . . . . . . . . 9 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
6448, 63imbi12d 333 . . . . . . . 8 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → ((∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))))
6564ex 449 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → (𝑣 = 𝑧 → ((∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))))
6613, 38, 65cbvald 2422 . . . . . 6 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → (∀𝑣(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∀𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))))
679, 66albid 2237 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → (∀𝑦𝑣(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))))
685, 67exbid 2238 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → (∃𝑥𝑦𝑣(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))))
691, 68mpbii 223 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
70693exp 1113 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (¬ ∀𝑧 𝑧 = 𝑤 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))))
71 axacndlem2 9642 . . 3 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
7271aecoms 2454 . 2 (∀𝑧 𝑧 = 𝑥 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
73 axacndlem3 9643 . . 3 (∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
7473aecoms 2454 . 2 (∀𝑧 𝑧 = 𝑦 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
75 nfae 2458 . . . 4 𝑦𝑧 𝑧 = 𝑤
76 simpr 479 . . . . . . 7 ((𝑦𝑧𝑧𝑤) → 𝑧𝑤)
7776alimi 1888 . . . . . 6 (∀𝑥(𝑦𝑧𝑧𝑤) → ∀𝑥 𝑧𝑤)
78 nd3 9623 . . . . . . 7 (∀𝑧 𝑧 = 𝑤 → ¬ ∀𝑥 𝑧𝑤)
7978pm2.21d 118 . . . . . 6 (∀𝑧 𝑧 = 𝑤 → (∀𝑥 𝑧𝑤 → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8077, 79syl5 34 . . . . 5 (∀𝑧 𝑧 = 𝑤 → (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8180axc4i 2278 . . . 4 (∀𝑧 𝑧 = 𝑤 → ∀𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8275, 81alrimi 2229 . . 3 (∀𝑧 𝑧 = 𝑤 → ∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
83 19.8a 2199 . . 3 (∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8482, 83syl 17 . 2 (∀𝑧 𝑧 = 𝑤 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8570, 72, 74, 84pm2.61iii 179 1 𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072  wal 1630  wex 1853  wnfc 2889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-reg 8664  ax-ac 9493
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-eprel 5179  df-fr 5225
This theorem is referenced by:  zfcndac  9653  axacprim  31912
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