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Theorem ac2 9166
Description: Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single explicit conjunction. (If you want to figure it out, the rewritten equivalent ac3 9167 is easier to understand.) Note: aceq0 8824 shows the logical equivalence to ax-ac 9164. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
Assertion
Ref Expression
ac2 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Proof of Theorem ac2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 ax-ac 9164 . 2 𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2 aceq0 8824 . 2 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)))
31, 2mpbir 220 1 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473  wex 1695  wral 2896  wrex 2897  ∃!wreu 2898
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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-ac 9164
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-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903
This theorem is referenced by:  ac3  9167
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