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Theorem axinfnd 9372
 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 9371 . . . . . . 7 (∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
21ax-gen 1719 . . . . . 6 𝑤(∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
3 nfnae 2317 . . . . . . . 8 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 nfnae 2317 . . . . . . . 8 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
53, 4nfan 1825 . . . . . . 7 𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
6 nfnae 2317 . . . . . . . . . 10 𝑥 ¬ ∀𝑦 𝑦 = 𝑥
7 nfnae 2317 . . . . . . . . . 10 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
86, 7nfan 1825 . . . . . . . . 9 𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
9 nfcvd 2762 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑤)
10 nfcvf 2784 . . . . . . . . . . 11 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1110adantl 482 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑧)
129, 11nfeld 2769 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤𝑧)
138, 12nfald 2162 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥 𝑤𝑧)
14 nfcvf 2784 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
1514adantr 481 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑥)
169, 15nfeld 2769 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤𝑥)
17 nfnae 2317 . . . . . . . . . . . 12 𝑤 ¬ ∀𝑦 𝑦 = 𝑥
18 nfnae 2317 . . . . . . . . . . . 12 𝑤 ¬ ∀𝑦 𝑦 = 𝑧
1917, 18nfan 1825 . . . . . . . . . . 11 𝑤(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
20 nfnae 2317 . . . . . . . . . . . . . 14 𝑧 ¬ ∀𝑦 𝑦 = 𝑥
21 nfnae 2317 . . . . . . . . . . . . . 14 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
2220, 21nfan 1825 . . . . . . . . . . . . 13 𝑧(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
2311, 15nfeld 2769 . . . . . . . . . . . . . 14 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑧𝑥)
2412, 23nfand 1823 . . . . . . . . . . . . 13 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑧𝑧𝑥))
2522, 24nfexd 2164 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧(𝑤𝑧𝑧𝑥))
2616, 25nfimd 1820 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2719, 26nfald 2162 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2816, 27nfand 1823 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
298, 28nfexd 2164 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))))
3013, 29nfimd 1820 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∀𝑥 𝑤𝑧 → ∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))))
31 nfcvd 2762 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑤)
32 nfcvf2 2785 . . . . . . . . . . . . 13 (¬ ∀𝑦 𝑦 = 𝑥𝑥𝑦)
3332adantr 481 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑦)
3431, 33nfeqd 2768 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥 𝑤 = 𝑦)
358, 34nfan1 2066 . . . . . . . . . 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 2683 . . . . . . . . . 10 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤𝑧𝑦𝑧))
3835, 37albid 2088 . . . . . . . . 9 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑥 𝑤𝑧 ↔ ∀𝑥 𝑦𝑧))
3936eleq1d 2683 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤𝑥𝑦𝑥))
40 nfcvd 2762 . . . . . . . . . . . . . . . . . 18 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑤)
41 nfcvf2 2785 . . . . . . . . . . . . . . . . . . 19 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
4241adantl 482 . . . . . . . . . . . . . . . . . 18 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑦)
4340, 42nfeqd 2768 . . . . . . . . . . . . . . . . 17 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑦)
4422, 43nfan1 2066 . . . . . . . . . . . . . . . 16 𝑧((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
4537anbi1d 740 . . . . . . . . . . . . . . . 16 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑧𝑧𝑥) ↔ (𝑦𝑧𝑧𝑥)))
4644, 45exbid 2089 . . . . . . . . . . . . . . 15 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑧(𝑤𝑧𝑧𝑥) ↔ ∃𝑧(𝑦𝑧𝑧𝑥)))
4739, 46imbi12d 334 . . . . . . . . . . . . . 14 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
4847ex 450 . . . . . . . . . . . . 13 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
495, 26, 48cbvald 2276 . . . . . . . . . . . 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 746 . . . . . . . . . 10 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5235, 51exbid 2089 . . . . . . . . 9 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑥(𝑤𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5338, 52imbi12d 334 . . . . . . . 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 450 . . . . . . 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 2276 . . . . . 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 223 . . . . 5 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
575619.21bi 2057 . . . 4 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5857ex 450 . . 3 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))))
59 nd1 9353 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
6059aecoms 2311 . . . 4 (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑦𝑧)
6160pm2.21d 118 . . 3 (∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
62 nd3 9355 . . . 4 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑥 𝑦𝑧)
6362pm2.21d 118 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
6458, 61, 63pm2.61ii 177 . 2 (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
656419.35ri 1804 1 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478  ∃wex 1701  Ⅎwnfc 2748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-reg 8441  ax-inf 8479 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-pr 4151 This theorem is referenced by:  zfcndinf  9384  axinfprim  31288
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