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Theorem axinfndlem1 10019
Description: Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
Assertion
Ref Expression
axinfndlem1 (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
Distinct variable group:   𝑦,𝑧

Proof of Theorem axinfndlem1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 zfinf 9094 . . . . 5 𝑤(𝑦𝑤 ∧ ∀𝑦(𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤)))
2 nfnae 2449 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3 nfnae 2449 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
42, 3nfan 1893 . . . . . 6 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
5 nfcvf 3005 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
65adantr 483 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑦)
7 nfcvd 2976 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑤)
86, 7nfeld 2987 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦𝑤)
9 nfnae 2449 . . . . . . . . 9 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
10 nfnae 2449 . . . . . . . . 9 𝑦 ¬ ∀𝑥 𝑥 = 𝑧
119, 10nfan 1893 . . . . . . . 8 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
12 nfnae 2449 . . . . . . . . . . 11 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
13 nfnae 2449 . . . . . . . . . . 11 𝑧 ¬ ∀𝑥 𝑥 = 𝑧
1412, 13nfan 1893 . . . . . . . . . 10 𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
15 nfcvf 3005 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
1615adantl 484 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑧)
176, 16nfeld 2987 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦𝑧)
1816, 7nfeld 2987 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧𝑤)
1917, 18nfand 1891 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑦𝑧𝑧𝑤))
2014, 19nfexd 2341 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑧(𝑦𝑧𝑧𝑤))
218, 20nfimd 1888 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤)))
2211, 21nfald 2340 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑦(𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤)))
238, 22nfand 1891 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑦𝑤 ∧ ∀𝑦(𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤))))
24 simpr 487 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → 𝑤 = 𝑥)
2524eleq2d 2896 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑦𝑤𝑦𝑥))
26 nfcvd 2976 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑦𝑤)
27 nfcvf2 3006 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
2827adantr 483 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑦𝑥)
2926, 28nfeqd 2986 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑦 𝑤 = 𝑥)
3011, 29nfan1 2192 . . . . . . . . 9 𝑦((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
31 nfcvd 2976 . . . . . . . . . . . . 13 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑧𝑤)
32 nfcvf2 3006 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑧𝑧𝑥)
3332adantl 484 . . . . . . . . . . . . 13 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑧𝑥)
3431, 33nfeqd 2986 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑥)
3514, 34nfan1 2192 . . . . . . . . . . 11 𝑧((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
36 elequ2 2122 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
3736anbi2d 630 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ((𝑦𝑧𝑧𝑤) ↔ (𝑦𝑧𝑧𝑥)))
3837adantl 484 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑦𝑧𝑧𝑤) ↔ (𝑦𝑧𝑧𝑥)))
3935, 38exbid 2217 . . . . . . . . . 10 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑧(𝑦𝑧𝑧𝑤) ↔ ∃𝑧(𝑦𝑧𝑧𝑥)))
4025, 39imbi12d 347 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
4130, 40albid 2216 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑦(𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤)) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
4225, 41anbi12d 632 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑦𝑤 ∧ ∀𝑦(𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
4342ex 415 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑦𝑤 ∧ ∀𝑦(𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))))
444, 23, 43cbvexd 2422 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤(𝑦𝑤 ∧ ∀𝑦(𝑦𝑤 → ∃𝑧(𝑦𝑧𝑧𝑤))) ↔ ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
451, 44mpbii 235 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
4645a1d 25 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
4746ex 415 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))))
48 nd1 10001 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
4948pm2.21d 121 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
50 nd2 10002 . . 3 (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑥 𝑦𝑧)
5150pm2.21d 121 . 2 (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))))
5247, 49, 51pm2.61ii 185 1 (∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1528  wex 1773  wnfc 2959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2383  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-reg 9048  ax-inf 9093
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-v 3495  df-dif 3937  df-un 3939  df-nul 4290  df-sn 4560  df-pr 4562
This theorem is referenced by:  axinfnd  10020
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