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Theorem axregnd 10077
 Description: A version of the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) (New usage is discouraged.)
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 10076 . . . 4 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)))
2 nfnae 2445 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑥
3 nfnae 2445 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑦
42, 3nfan 1900 . . . . 5 𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
5 nfnae 2445 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
6 nfnae 2445 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
75, 6nfan 1900 . . . . . . 7 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
8 nfcvf 2945 . . . . . . . . . 10 (¬ ∀𝑧 𝑧 = 𝑥𝑧𝑥)
98nfcrd 2908 . . . . . . . . 9 (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑤𝑥)
109adantr 484 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤𝑥)
11 nfcvf 2945 . . . . . . . . . . 11 (¬ ∀𝑧 𝑧 = 𝑦𝑧𝑦)
1211nfcrd 2908 . . . . . . . . . 10 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑤𝑦)
1312nfnd 1859 . . . . . . . . 9 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 ¬ 𝑤𝑦)
1413adantl 485 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 ¬ 𝑤𝑦)
1510, 14nfimd 1895 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤𝑥 → ¬ 𝑤𝑦))
16 elequ1 2118 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
17 elequ1 2118 . . . . . . . . . 10 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1817notbid 321 . . . . . . . . 9 (𝑤 = 𝑧 → (¬ 𝑤𝑦 ↔ ¬ 𝑧𝑦))
1916, 18imbi12d 348 . . . . . . . 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 2417 . . . . . 6 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦) ↔ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2221anbi2d 631 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
234, 22exbid 2223 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑥(𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
241, 23syl5ib 247 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
2524ex 416 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
26 axregndlem1 10075 . . 3 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
2726aecoms 2439 . 2 (∀𝑧 𝑧 = 𝑥 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
28 19.8a 2178 . . 3 (𝑥𝑦 → ∃𝑥 𝑥𝑦)
29 nfae 2444 . . . 4 𝑥𝑧 𝑧 = 𝑦
30 elirrv 9106 . . . . . . . . 9 ¬ 𝑧𝑧
31 elequ2 2126 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑧𝑦))
3230, 31mtbii 329 . . . . . . . 8 (𝑧 = 𝑦 → ¬ 𝑧𝑦)
3332a1d 25 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥 → ¬ 𝑧𝑦))
3433alimi 1813 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))
3534anim2i 619 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧 𝑧 = 𝑦) → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
3635expcom 417 . . . 4 (∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3729, 36eximd 2214 . . 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 186 1 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-reg 9102 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-nul 4228  df-sn 4526  df-pr 4528 This theorem is referenced by:  zfcndreg  10090  axregprim  33175
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