MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axacndlem2 Structured version   Visualization version   GIF version

Theorem axacndlem2 9415
Description: Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
Assertion
Ref Expression
axacndlem2 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Proof of Theorem axacndlem2
StepHypRef Expression
1 nfae 2314 . . 3 𝑦𝑥 𝑥 = 𝑧
2 nfae 2314 . . . 4 𝑧𝑥 𝑥 = 𝑧
3 simpr 477 . . . . . 6 ((𝑦𝑧𝑧𝑤) → 𝑧𝑤)
43alimi 1737 . . . . 5 (∀𝑥(𝑦𝑧𝑧𝑤) → ∀𝑥 𝑧𝑤)
5 nd1 9394 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑥 𝑧𝑤)
65pm2.21d 118 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑧𝑤 → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
74, 6syl5 34 . . . 4 (∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
82, 7alrimi 2080 . . 3 (∀𝑥 𝑥 = 𝑧 → ∀𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
91, 8alrimi 2080 . 2 (∀𝑥 𝑥 = 𝑧 → ∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
10 19.8a 2050 . 2 (∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
119, 10syl 17 1 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-reg 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-un 3572  df-nul 3908  df-sn 4169  df-pr 4171
This theorem is referenced by:  axacndlem4  9417  axacnd  9419
  Copyright terms: Public domain W3C validator