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

Theorem axacndlem2 10648
Description: Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.)
Assertion
Ref Expression
axacndlem2 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Proof of Theorem axacndlem2
StepHypRef Expression
1 nfae 2438 . . 3 𝑦𝑥 𝑥 = 𝑧
2 nfae 2438 . . . 4 𝑧𝑥 𝑥 = 𝑧
3 simpr 484 . . . . . 6 ((𝑦𝑧𝑧𝑤) → 𝑧𝑤)
43alimi 1811 . . . . 5 (∀𝑥(𝑦𝑧𝑧𝑤) → ∀𝑥 𝑧𝑤)
5 nd1 10627 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑥 𝑧𝑤)
65pm2.21d 121 . . . . 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 2213 . . 3 (∀𝑥 𝑥 = 𝑧 → ∀𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
91, 8alrimi 2213 . 2 (∀𝑥 𝑥 = 𝑧 → ∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
10919.8ad 2182 1 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708  ax-sep 5296  ax-pr 5432  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  axacndlem4  10650  axacnd  10652
  Copyright terms: Public domain W3C validator