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Theorem axregnd 10015
Description: A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
Assertion
Ref Expression
axregnd (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Proof of Theorem axregnd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axregndlem2 10014 . . . 4 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)))
2 nfnae 2453 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑥
3 nfnae 2453 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑦
42, 3nfan 1893 . . . . 5 𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
5 nfnae 2453 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
6 nfnae 2453 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
75, 6nfan 1893 . . . . . . 7 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
8 nfcvf 3012 . . . . . . . . . 10 (¬ ∀𝑧 𝑧 = 𝑥𝑧𝑥)
98nfcrd 2974 . . . . . . . . 9 (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑤𝑥)
109adantr 481 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤𝑥)
11 nfcvf 3012 . . . . . . . . . . 11 (¬ ∀𝑧 𝑧 = 𝑦𝑧𝑦)
1211nfcrd 2974 . . . . . . . . . 10 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑤𝑦)
1312nfnd 1851 . . . . . . . . 9 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 ¬ 𝑤𝑦)
1413adantl 482 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 ¬ 𝑤𝑦)
1510, 14nfimd 1888 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤𝑥 → ¬ 𝑤𝑦))
16 elequ1 2114 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
17 elequ1 2114 . . . . . . . . . 10 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1817notbid 319 . . . . . . . . 9 (𝑤 = 𝑧 → (¬ 𝑤𝑦 ↔ ¬ 𝑧𝑦))
1916, 18imbi12d 346 . . . . . . . 8 (𝑤 = 𝑧 → ((𝑤𝑥 → ¬ 𝑤𝑦) ↔ (𝑧𝑥 → ¬ 𝑧𝑦)))
2019a1i 11 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑤 = 𝑧 → ((𝑤𝑥 → ¬ 𝑤𝑦) ↔ (𝑧𝑥 → ¬ 𝑧𝑦))))
217, 15, 20cbvald 2424 . . . . . 6 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦) ↔ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2221anbi2d 628 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
234, 22exbid 2218 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑥(𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
241, 23syl5ib 245 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
2524ex 413 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
26 axregndlem1 10013 . . 3 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
2726aecoms 2447 . 2 (∀𝑧 𝑧 = 𝑥 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
28 19.8a 2172 . . 3 (𝑥𝑦 → ∃𝑥 𝑥𝑦)
29 nfae 2452 . . . 4 𝑥𝑧 𝑧 = 𝑦
30 elirrv 9049 . . . . . . . . 9 ¬ 𝑧𝑧
31 elequ2 2122 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑧𝑦))
3230, 31mtbii 327 . . . . . . . 8 (𝑧 = 𝑦 → ¬ 𝑧𝑦)
3332a1d 25 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥 → ¬ 𝑧𝑦))
3433alimi 1805 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))
3534anim2i 616 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧 𝑧 = 𝑦) → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
3635expcom 414 . . . 4 (∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3729, 36eximd 2209 . . 3 (∀𝑧 𝑧 = 𝑦 → (∃𝑥 𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3828, 37syl5 34 . 2 (∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3925, 27, 38pm2.61ii 184 1 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1528  wex 1773  wnf 1777
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 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-reg 9045
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-v 3502  df-dif 3943  df-un 3945  df-nul 4296  df-sn 4565  df-pr 4567
This theorem is referenced by:  zfcndreg  10028  axregprim  32818
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