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Theorem axacndlem1 9825
Description: Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
Assertion
Ref Expression
axacndlem1 (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Proof of Theorem axacndlem1
StepHypRef Expression
1 nfae 2370 . . 3 𝑦𝑥 𝑥 = 𝑦
2 nfae 2370 . . . 4 𝑧𝑥 𝑥 = 𝑦
3 simpl 475 . . . . . 6 ((𝑦𝑧𝑧𝑤) → 𝑦𝑧)
43alimi 1775 . . . . 5 (∀𝑥(𝑦𝑧𝑧𝑤) → ∀𝑥 𝑦𝑧)
5 nd1 9805 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
65pm2.21d 119 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦𝑧 → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
74, 6syl5 34 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
82, 7alrimi 2144 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
91, 8alrimi 2144 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
10919.8ad 2111 1 (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1506  wex 1743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-reg 8849
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-v 3410  df-dif 3825  df-un 3827  df-nul 4173  df-sn 4436  df-pr 4438
This theorem is referenced by:  axacndlem4  9828  axacndlem5  9829
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