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Theorem aceq2 8825
 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 2901 . . . . 5 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 1889 . . . . 5 (∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitri 263 . . . 4 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
4 biidd 251 . . . . 5 (𝑤 = 𝑡 → (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
54cbvralv 3147 . . . 4 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
6 n0 3890 . . . . 5 (𝑧 ≠ ∅ ↔ ∃𝑡 𝑡𝑧)
7 eleq2 2677 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑧𝑣𝑧𝑢))
8 eleq2 2677 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑤𝑣𝑤𝑢))
97, 8anbi12d 743 . . . . . . . 8 (𝑣 = 𝑢 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑢𝑤𝑢)))
109cbvrexv 3148 . . . . . . 7 (∃𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃𝑢𝑦 (𝑧𝑢𝑤𝑢))
1110reubii 3105 . . . . . 6 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢))
12 eleq1 2676 . . . . . . . . 9 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
1312anbi2d 736 . . . . . . . 8 (𝑤 = 𝑣 → ((𝑧𝑢𝑤𝑢) ↔ (𝑧𝑢𝑣𝑢)))
1413rexbidv 3034 . . . . . . 7 (𝑤 = 𝑣 → (∃𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
1514cbvreuv 3149 . . . . . 6 (∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
1611, 15bitri 263 . . . . 5 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
176, 16imbi12i 339 . . . 4 ((𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
183, 5, 173bitr4i 291 . . 3 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
1918ralbii 2963 . 2 (∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
2019exbii 1764 1 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by:  dfac7  8837  ac3  9167
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