MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axregndlem2 Structured version   Visualization version   GIF version

Theorem axregndlem2 9463
Description: Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
axregndlem2 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Distinct variable group:   𝑦,𝑧

Proof of Theorem axregndlem2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axreg2 8539 . . . . . 6 (𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
21ax-gen 1762 . . . . 5 𝑤(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
3 nfnae 2351 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4 nfnae 2351 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
53, 4nfan 1868 . . . . . 6 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
6 nfcvd 2794 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑤)
7 nfcvf 2817 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
87adantr 480 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑦)
96, 8nfeld 2802 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤𝑦)
10 nfv 1883 . . . . . . . 8 𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
11 nfnae 2351 . . . . . . . . . . 11 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
12 nfnae 2351 . . . . . . . . . . 11 𝑧 ¬ ∀𝑥 𝑥 = 𝑧
1311, 12nfan 1868 . . . . . . . . . 10 𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
14 nfcvf 2817 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
1514adantl 481 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑧)
1615, 6nfeld 2802 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧𝑤)
1715, 8nfeld 2802 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧𝑦)
1817nfnd 1825 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 ¬ 𝑧𝑦)
1916, 18nfimd 1863 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧𝑤 → ¬ 𝑧𝑦))
2013, 19nfald 2201 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑧(𝑧𝑤 → ¬ 𝑧𝑦))
219, 20nfand 1866 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
2210, 21nfexd 2203 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
239, 22nfimd 1863 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))))
24 simpr 476 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → 𝑤 = 𝑥)
2524eleq1d 2715 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑤𝑦𝑥𝑦))
26 nfcvd 2794 . . . . . . . . . . . . . . 15 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑧𝑤)
27 nfcvf2 2818 . . . . . . . . . . . . . . . 16 (¬ ∀𝑥 𝑥 = 𝑧𝑧𝑥)
2827adantl 481 . . . . . . . . . . . . . . 15 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑧𝑥)
2926, 28nfeqd 2801 . . . . . . . . . . . . . 14 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑥)
3013, 29nfan1 2106 . . . . . . . . . . . . 13 𝑧((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
3124eleq2d 2716 . . . . . . . . . . . . . 14 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑧𝑤𝑧𝑥))
3231imbi1d 330 . . . . . . . . . . . . 13 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑧𝑤 → ¬ 𝑧𝑦) ↔ (𝑧𝑥 → ¬ 𝑧𝑦)))
3330, 32albid 2128 . . . . . . . . . . . 12 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦) ↔ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
3425, 33anbi12d 747 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3534ex 449 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
365, 21, 35cbvexd 2314 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3736adantr 480 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3825, 37imbi12d 333 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
3938ex 449 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))))
405, 23, 39cbvald 2313 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑤(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ ∀𝑥(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
412, 40mpbii 223 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∀𝑥(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
424119.21bi 2097 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
4342ex 449 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
44 elirrv 8542 . . . . 5 ¬ 𝑥𝑥
45 elequ2 2044 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
4644, 45mtbii 315 . . . 4 (𝑥 = 𝑦 → ¬ 𝑥𝑦)
4746sps 2093 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
4847pm2.21d 118 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
49 axregndlem1 9462 . 2 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
5043, 48, 49pm2.61ii 177 1 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wex 1744  wnfc 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by:  axregnd  9464
  Copyright terms: Public domain W3C validator