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Theorem axacndlem3 9632
Description: Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
Assertion
Ref Expression
axacndlem3 (∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Proof of Theorem axacndlem3
StepHypRef Expression
1 nfae 2467 . . . 4 𝑧𝑦 𝑦 = 𝑧
2 simpl 468 . . . . . 6 ((𝑦𝑧𝑧𝑤) → 𝑦𝑧)
32alimi 1886 . . . . 5 (∀𝑥(𝑦𝑧𝑧𝑤) → ∀𝑥 𝑦𝑧)
4 nd3 9612 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑥 𝑦𝑧)
54pm2.21d 119 . . . . 5 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦𝑧 → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
63, 5syl5 34 . . . 4 (∀𝑦 𝑦 = 𝑧 → (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
71, 6alrimi 2237 . . 3 (∀𝑦 𝑦 = 𝑧 → ∀𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
87axc4i 2294 . 2 (∀𝑦 𝑦 = 𝑧 → ∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
9 19.8a 2205 . 2 (∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
108, 9syl 17 1 (∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-reg 8652
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-dif 3724  df-un 3726  df-nul 4062  df-sn 4315  df-pr 4317
This theorem is referenced by:  axacndlem5  9634  axacnd  9635
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