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Theorem aceq2 8886
Description: Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
aceq2 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Proof of Theorem aceq2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 df-ral 2912 . . . . 5 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 1899 . . . . 5 (∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitri 264 . . . 4 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
4 biidd 252 . . . . 5 (𝑤 = 𝑡 → (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
54cbvralv 3159 . . . 4 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
6 n0 3907 . . . . 5 (𝑧 ≠ ∅ ↔ ∃𝑡 𝑡𝑧)
7 elequ2 2001 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑧𝑣𝑧𝑢))
8 elequ2 2001 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑤𝑣𝑤𝑢))
97, 8anbi12d 746 . . . . . . . 8 (𝑣 = 𝑢 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑢𝑤𝑢)))
109cbvrexv 3160 . . . . . . 7 (∃𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃𝑢𝑦 (𝑧𝑢𝑤𝑢))
1110reubii 3117 . . . . . 6 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢))
12 eleq1 2686 . . . . . . . . 9 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
1312anbi2d 739 . . . . . . . 8 (𝑤 = 𝑣 → ((𝑧𝑢𝑤𝑢) ↔ (𝑧𝑢𝑣𝑢)))
1413rexbidv 3045 . . . . . . 7 (𝑤 = 𝑣 → (∃𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
1514cbvreuv 3161 . . . . . 6 (∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
1611, 15bitri 264 . . . . 5 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
176, 16imbi12i 340 . . . 4 ((𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
183, 5, 173bitr4i 292 . . 3 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
1918ralbii 2974 . 2 (∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
2019exbii 1771 1 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701  wne 2790  wral 2907  wrex 2908  ∃!wreu 2909  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-v 3188  df-dif 3558  df-nul 3892
This theorem is referenced by:  dfac7  8898  ac3  9228
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