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Theorem aceq0 9192
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 9534. (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 9191 . 2 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
2 equequ2 2123 . . . . . . . . . 10 (𝑣 = 𝑥 → (𝑢 = 𝑣𝑢 = 𝑥))
32bibi2d 333 . . . . . . . . 9 (𝑣 = 𝑥 → ((∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ (∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑥)))
4 elequ2 2169 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝑤𝑡𝑤𝑥))
54anbi2d 622 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((𝑢𝑤𝑤𝑡) ↔ (𝑢𝑤𝑤𝑥)))
6 elequ2 2169 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝑢𝑡𝑢𝑥))
7 elequ1 2162 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝑡𝑦𝑥𝑦))
86, 7anbi12d 624 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((𝑢𝑡𝑡𝑦) ↔ (𝑢𝑥𝑥𝑦)))
95, 8anbi12d 624 . . . . . . . . . . 11 (𝑡 = 𝑥 → (((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ ((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦))))
109cbvexvw 2137 . . . . . . . . . 10 (∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ ∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)))
1110bibi1i 329 . . . . . . . . 9 ((∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑥) ↔ (∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥))
123, 11syl6bb 278 . . . . . . . 8 (𝑣 = 𝑥 → ((∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ (∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥)))
1312albidv 2015 . . . . . . 7 (𝑣 = 𝑥 → (∀𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ ∀𝑢(∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥)))
14 elequ1 2162 . . . . . . . . . . . 12 (𝑢 = 𝑧 → (𝑢𝑤𝑧𝑤))
1514anbi1d 623 . . . . . . . . . . 11 (𝑢 = 𝑧 → ((𝑢𝑤𝑤𝑥) ↔ (𝑧𝑤𝑤𝑥)))
16 elequ1 2162 . . . . . . . . . . . 12 (𝑢 = 𝑧 → (𝑢𝑥𝑧𝑥))
1716anbi1d 623 . . . . . . . . . . 11 (𝑢 = 𝑧 → ((𝑢𝑥𝑥𝑦) ↔ (𝑧𝑥𝑥𝑦)))
1815, 17anbi12d 624 . . . . . . . . . 10 (𝑢 = 𝑧 → (((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ ((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦))))
1918exbidv 2016 . . . . . . . . 9 (𝑢 = 𝑧 → (∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ ∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦))))
20 equequ1 2122 . . . . . . . . 9 (𝑢 = 𝑧 → (𝑢 = 𝑥𝑧 = 𝑥))
2119, 20bibi12d 336 . . . . . . . 8 (𝑢 = 𝑧 → ((∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥) ↔ (∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
2221cbvalvw 2136 . . . . . . 7 (∀𝑢(∃𝑥((𝑢𝑤𝑤𝑥) ∧ (𝑢𝑥𝑥𝑦)) ↔ 𝑢 = 𝑥) ↔ ∀𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥))
2313, 22syl6bb 278 . . . . . 6 (𝑣 = 𝑥 → (∀𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ ∀𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
2423cbvexvw 2137 . . . . 5 (∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣) ↔ ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥))
2524imbi2i 327 . . . 4 (((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)) ↔ ((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
26252albii 1915 . . 3 (∀𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)) ↔ ∀𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
2726exbii 1943 . 2 (∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
281, 27bitr4i 269 1 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wex 1874  wral 3055  wrex 3056  ∃!wreu 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-reu 3062
This theorem is referenced by:  dfac0  9208  ac2  9536
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