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Theorem aceq2 10152
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 3052 . . . . 5 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 1938 . . . . 5 (∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitri 274 . . . 4 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
4 biidd 261 . . . . 5 (𝑤 = 𝑡 → (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
54cbvralvw 3225 . . . 4 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
6 n0 4346 . . . . 5 (𝑧 ≠ ∅ ↔ ∃𝑡 𝑡𝑧)
7 elequ2 2114 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑧𝑣𝑧𝑢))
8 elequ2 2114 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑤𝑣𝑤𝑢))
97, 8anbi12d 630 . . . . . . . 8 (𝑣 = 𝑢 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑢𝑤𝑢)))
109cbvrexvw 3226 . . . . . . 7 (∃𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃𝑢𝑦 (𝑧𝑢𝑤𝑢))
1110reubii 3373 . . . . . 6 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢))
12 elequ1 2106 . . . . . . . . 9 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
1312anbi2d 628 . . . . . . . 8 (𝑤 = 𝑣 → ((𝑧𝑢𝑤𝑢) ↔ (𝑧𝑢𝑣𝑢)))
1413rexbidv 3169 . . . . . . 7 (𝑤 = 𝑣 → (∃𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
1514cbvreuvw 3388 . . . . . 6 (∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
1611, 15bitri 274 . . . . 5 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
176, 16imbi12i 349 . . . 4 ((𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
183, 5, 173bitr4i 302 . . 3 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
1918ralbii 3083 . 2 (∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
2019exbii 1843 1 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1532  wex 1774  wne 2930  wral 3051  wrex 3060  ∃!wreu 3362  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-dif 3949  df-nul 4323
This theorem is referenced by:  dfac7  10165  ac3  10493
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