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Theorem axinfnd 9826
Description: A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
Assertion
Ref Expression
axinfnd 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))

Proof of Theorem axinfnd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axinfndlem1 9825 . . . . . . 7 (∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
21ax-gen 1758 . . . . . 6 𝑤(∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
3 nfnae 2370 . . . . . . . 8 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 nfnae 2370 . . . . . . . 8 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
53, 4nfan 1862 . . . . . . 7 𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
6 nfnae 2370 . . . . . . . . . 10 𝑥 ¬ ∀𝑦 𝑦 = 𝑥
7 nfnae 2370 . . . . . . . . . 10 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
86, 7nfan 1862 . . . . . . . . 9 𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
9 nfcvd 2933 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑤)
10 nfcvf 2958 . . . . . . . . . . 11 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1110adantl 474 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑧)
129, 11nfeld 2941 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤𝑧)
138, 12nfald 2268 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥 𝑤𝑧)
14 nfcvf 2958 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
1514adantr 473 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑥)
169, 15nfeld 2941 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤𝑥)
17 nfnae 2370 . . . . . . . . . . . 12 𝑤 ¬ ∀𝑦 𝑦 = 𝑥
18 nfnae 2370 . . . . . . . . . . . 12 𝑤 ¬ ∀𝑦 𝑦 = 𝑧
1917, 18nfan 1862 . . . . . . . . . . 11 𝑤(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
20 nfnae 2370 . . . . . . . . . . . . . 14 𝑧 ¬ ∀𝑦 𝑦 = 𝑥
21 nfnae 2370 . . . . . . . . . . . . . 14 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
2220, 21nfan 1862 . . . . . . . . . . . . 13 𝑧(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
2311, 15nfeld 2941 . . . . . . . . . . . . . 14 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑧𝑥)
2412, 23nfand 1860 . . . . . . . . . . . . 13 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑧𝑧𝑥))
2522, 24nfexd 2269 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧(𝑤𝑧𝑧𝑥))
2616, 25nfimd 1857 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2719, 26nfald 2268 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2816, 27nfand 1860 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
298, 28nfexd 2269 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
3013, 29nfimd 1857 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))))
31 nfcvd 2933 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑤)
32 nfcvf2 2959 . . . . . . . . . . . . 13 (¬ ∀𝑦 𝑦 = 𝑥𝑥𝑦)
3332adantr 473 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑦)
3431, 33nfeqd 2940 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥 𝑤 = 𝑦)
358, 34nfan1 2129 . . . . . . . . . 10 𝑥((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
36 simpr 477 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → 𝑤 = 𝑦)
3736eleq1d 2850 . . . . . . . . . 10 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤𝑧𝑦𝑧))
3835, 37albid 2154 . . . . . . . . 9 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑥 𝑤𝑧 ↔ ∀𝑥 𝑦𝑧))
3936eleq1d 2850 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤𝑥𝑦𝑥))
40 nfcvd 2933 . . . . . . . . . . . . . . . . . 18 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑤)
41 nfcvf2 2959 . . . . . . . . . . . . . . . . . . 19 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
4241adantl 474 . . . . . . . . . . . . . . . . . 18 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑦)
4340, 42nfeqd 2940 . . . . . . . . . . . . . . . . 17 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑦)
4422, 43nfan1 2129 . . . . . . . . . . . . . . . 16 𝑧((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
4537anbi1d 620 . . . . . . . . . . . . . . . 16 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑧𝑧𝑥) ↔ (𝑦𝑧𝑧𝑥)))
4644, 45exbid 2155 . . . . . . . . . . . . . . 15 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑧(𝑤𝑧𝑧𝑥) ↔ ∃𝑧(𝑦𝑧𝑧𝑥)))
4739, 46imbi12d 337 . . . . . . . . . . . . . 14 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
4847ex 405 . . . . . . . . . . . . 13 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
495, 26, 48cbvald 2341 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
5049adantr 473 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
5139, 50anbi12d 621 . . . . . . . . . 10 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5235, 51exbid 2155 . . . . . . . . 9 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5338, 52imbi12d 337 . . . . . . . 8 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))) ↔ (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))))
5453ex 405 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))) ↔ (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))))
555, 30, 54cbvald 2341 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))) ↔ ∀𝑦(∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))))
562, 55mpbii 225 . . . . 5 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
575619.21bi 2117 . . . 4 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5857ex 405 . . 3 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))))
59 nd1 9807 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
6059aecoms 2364 . . . 4 (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑦𝑧)
6160pm2.21d 119 . . 3 (∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
62 nd3 9809 . . . 4 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑥 𝑦𝑧)
6362pm2.21d 119 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
6458, 61, 63pm2.61ii 178 . 2 (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
656419.35ri 1842 1 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wal 1505  wex 1742  wnfc 2916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-reg 8851  ax-inf 8895
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-v 3417  df-dif 3832  df-un 3834  df-nul 4179  df-sn 4442  df-pr 4444
This theorem is referenced by:  zfcndinf  9838  axinfprim  32458
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