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Theorem axregnd 9378
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 9377 . . . 4 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)))
2 nfnae 2317 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑥
3 nfnae 2317 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑦
42, 3nfan 1825 . . . . 5 𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
5 nfnae 2317 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
6 nfnae 2317 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
75, 6nfan 1825 . . . . . . 7 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
8 nfcvf 2784 . . . . . . . . . 10 (¬ ∀𝑧 𝑧 = 𝑥𝑧𝑥)
98nfcrd 2767 . . . . . . . . 9 (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑤𝑥)
109adantr 481 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤𝑥)
11 nfcvf 2784 . . . . . . . . . . 11 (¬ ∀𝑧 𝑧 = 𝑦𝑧𝑦)
1211nfcrd 2767 . . . . . . . . . 10 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑤𝑦)
1312nfnd 1782 . . . . . . . . 9 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 ¬ 𝑤𝑦)
1413adantl 482 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 ¬ 𝑤𝑦)
1510, 14nfimd 1820 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤𝑥 → ¬ 𝑤𝑦))
16 elequ1 1994 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
17 elequ1 1994 . . . . . . . . . 10 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1817notbid 308 . . . . . . . . 9 (𝑤 = 𝑧 → (¬ 𝑤𝑦 ↔ ¬ 𝑧𝑦))
1916, 18imbi12d 334 . . . . . . . 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 2276 . . . . . 6 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦) ↔ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2221anbi2d 739 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
234, 22exbid 2089 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑥(𝑥𝑦 ∧ ∀𝑤(𝑤𝑥 → ¬ 𝑤𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
241, 23syl5ib 234 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
2524ex 450 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
26 axregndlem1 9376 . . 3 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
2726aecoms 2311 . 2 (∀𝑧 𝑧 = 𝑥 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
28 19.8a 2049 . . 3 (𝑥𝑦 → ∃𝑥 𝑥𝑦)
29 nfae 2315 . . . 4 𝑥𝑧 𝑧 = 𝑦
30 elirrv 8456 . . . . . . . . 9 ¬ 𝑧𝑧
31 elequ2 2001 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑧𝑦))
3230, 31mtbii 316 . . . . . . . 8 (𝑧 = 𝑦 → ¬ 𝑧𝑦)
3332a1d 25 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥 → ¬ 𝑧𝑦))
3433alimi 1736 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))
3534anim2i 592 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧 𝑧 = 𝑦) → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
3635expcom 451 . . . 4 (∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3729, 36eximd 2083 . . 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 177 1 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478  wex 1701  wnf 1705
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 4746  ax-nul 4754  ax-pr 4872  ax-reg 8449
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 3191  df-dif 3562  df-un 3564  df-nul 3897  df-sn 4154  df-pr 4156
This theorem is referenced by:  zfcndreg  9391  axregprim  31325
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