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Theorem axinfnd 10016
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 10015 . . . . . . 7 (∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
21ax-gen 1787 . . . . . 6 𝑤(∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
3 nfnae 2448 . . . . . . . 8 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 nfnae 2448 . . . . . . . 8 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
53, 4nfan 1891 . . . . . . 7 𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
6 nfnae 2448 . . . . . . . . . 10 𝑥 ¬ ∀𝑦 𝑦 = 𝑥
7 nfnae 2448 . . . . . . . . . 10 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
86, 7nfan 1891 . . . . . . . . 9 𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
9 nfcvd 2975 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑤)
10 nfcvf 3004 . . . . . . . . . . 11 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1110adantl 482 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑧)
129, 11nfeld 2986 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤𝑧)
138, 12nfald 2338 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥 𝑤𝑧)
14 nfcvf 3004 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
1514adantr 481 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑥)
169, 15nfeld 2986 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤𝑥)
17 nfnae 2448 . . . . . . . . . . . 12 𝑤 ¬ ∀𝑦 𝑦 = 𝑥
18 nfnae 2448 . . . . . . . . . . . 12 𝑤 ¬ ∀𝑦 𝑦 = 𝑧
1917, 18nfan 1891 . . . . . . . . . . 11 𝑤(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
20 nfnae 2448 . . . . . . . . . . . . . 14 𝑧 ¬ ∀𝑦 𝑦 = 𝑥
21 nfnae 2448 . . . . . . . . . . . . . 14 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
2220, 21nfan 1891 . . . . . . . . . . . . 13 𝑧(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
2311, 15nfeld 2986 . . . . . . . . . . . . . 14 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑧𝑥)
2412, 23nfand 1889 . . . . . . . . . . . . 13 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑧𝑧𝑥))
2522, 24nfexd 2339 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧(𝑤𝑧𝑧𝑥))
2616, 25nfimd 1886 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2719, 26nfald 2338 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2816, 27nfand 1889 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
298, 28nfexd 2339 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
3013, 29nfimd 1886 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))))
31 nfcvd 2975 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑤)
32 nfcvf2 3005 . . . . . . . . . . . . 13 (¬ ∀𝑦 𝑦 = 𝑥𝑥𝑦)
3332adantr 481 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑦)
3431, 33nfeqd 2985 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥 𝑤 = 𝑦)
358, 34nfan1 2190 . . . . . . . . . 10 𝑥((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
36 simpr 485 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → 𝑤 = 𝑦)
3736eleq1d 2894 . . . . . . . . . 10 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤𝑧𝑦𝑧))
3835, 37albid 2214 . . . . . . . . 9 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑥 𝑤𝑧 ↔ ∀𝑥 𝑦𝑧))
3936eleq1d 2894 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤𝑥𝑦𝑥))
40 nfcvd 2975 . . . . . . . . . . . . . . . . . 18 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑤)
41 nfcvf2 3005 . . . . . . . . . . . . . . . . . . 19 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
4241adantl 482 . . . . . . . . . . . . . . . . . 18 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑦)
4340, 42nfeqd 2985 . . . . . . . . . . . . . . . . 17 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑦)
4422, 43nfan1 2190 . . . . . . . . . . . . . . . 16 𝑧((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
4537anbi1d 629 . . . . . . . . . . . . . . . 16 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑧𝑧𝑥) ↔ (𝑦𝑧𝑧𝑥)))
4644, 45exbid 2215 . . . . . . . . . . . . . . 15 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑧(𝑤𝑧𝑧𝑥) ↔ ∃𝑧(𝑦𝑧𝑧𝑥)))
4739, 46imbi12d 346 . . . . . . . . . . . . . 14 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
4847ex 413 . . . . . . . . . . . . 13 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
495, 26, 48cbvald 2419 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
5049adantr 481 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
5139, 50anbi12d 630 . . . . . . . . . 10 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5235, 51exbid 2215 . . . . . . . . 9 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5338, 52imbi12d 346 . . . . . . . 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 413 . . . . . . 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 2419 . . . . . 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 234 . . . . 5 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
575619.21bi 2178 . . . 4 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5857ex 413 . . 3 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))))
59 nd1 9997 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
6059aecoms 2442 . . . 4 (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑦𝑧)
6160pm2.21d 121 . . 3 (∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
62 nd3 9999 . . . 4 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑥 𝑦𝑧)
6362pm2.21d 121 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
6458, 61, 63pm2.61ii 184 . 2 (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
656419.35ri 1871 1 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526  wex 1771  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-reg 9044  ax-inf 9089
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-un 3938  df-nul 4289  df-sn 4558  df-pr 4560
This theorem is referenced by:  zfcndinf  10028  axinfprim  32829
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