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Theorem aceq0 9140
Description: Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 9482. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
aceq0 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡

Proof of Theorem aceq0
StepHypRef Expression
1 aceq1 9139 . 2 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
2 equequ2 2110 . . . . . . . . . 10 (𝑣 = 𝑥 → (𝑢 = 𝑣𝑢 = 𝑥))
32bibi2d 331 . . . . . . . . 9 (𝑣 = 𝑥 → ((∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ (∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑥)))
4 elequ2 2158 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝑤𝑡𝑤𝑥))
54anbi2d 606 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((𝑢𝑤𝑤𝑡) ↔ (𝑢𝑤𝑤𝑥)))
6 elequ2 2158 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝑢𝑡𝑢𝑥))
7 elequ1 2151 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝑡𝑦𝑥𝑦))
86, 7anbi12d 608 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((𝑢𝑡𝑡𝑦) ↔ (𝑢𝑥𝑥𝑦)))
95, 8anbi12d 608 . . . . . . . . . . 11 (𝑡 = 𝑥 → (((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ ((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦))))
109cbvexv 2434 . . . . . . . . . 10 (∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ ∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)))
1110bibi1i 327 . . . . . . . . 9 ((∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑥) ↔ (∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥))
123, 11syl6bb 276 . . . . . . . 8 (𝑣 = 𝑥 → ((∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ (∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥)))
1312albidv 2000 . . . . . . 7 (𝑣 = 𝑥 → (∀𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ ∀𝑢(∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥)))
14 elequ1 2151 . . . . . . . . . . . 12 (𝑢 = 𝑧 → (𝑢𝑤𝑧𝑤))
1514anbi1d 607 . . . . . . . . . . 11 (𝑢 = 𝑧 → ((𝑢𝑤𝑤𝑥) ↔ (𝑧𝑤𝑤𝑥)))
16 elequ1 2151 . . . . . . . . . . . 12 (𝑢 = 𝑧 → (𝑢𝑥𝑧𝑥))
1716anbi1d 607 . . . . . . . . . . 11 (𝑢 = 𝑧 → ((𝑢𝑥𝑥𝑦) ↔ (𝑧𝑥𝑥𝑦)))
1815, 17anbi12d 608 . . . . . . . . . 10 (𝑢 = 𝑧 → (((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ ((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦))))
1918exbidv 2001 . . . . . . . . 9 (𝑢 = 𝑧 → (∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ ∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦))))
20 equequ1 2109 . . . . . . . . 9 (𝑢 = 𝑧 → (𝑢 = 𝑥𝑧 = 𝑥))
2119, 20bibi12d 334 . . . . . . . 8 (𝑢 = 𝑧 → ((∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥) ↔ (∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
2221cbvalv 2433 . . . . . . 7 (∀𝑢(∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥) ↔ ∀𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥))
2313, 22syl6bb 276 . . . . . 6 (𝑣 = 𝑥 → (∀𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ ∀𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
2423cbvexv 2434 . . . . 5 (∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥))
2524imbi2i 325 . . . 4 (((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)) ↔ ((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
26252albii 1895 . . 3 (∀𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)) ↔ ∀𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
2726exbii 1923 . 2 (∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
281, 27bitr4i 267 1 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1628  wex 1851  wral 3060  wrex 3061  ∃!wreu 3062
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-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
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-eu 2621  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-reu 3067
This theorem is referenced by:  dfac0  9156  ac2  9484
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