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Theorem ac2 8103
 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 8104 is easier to understand.) Note: aceq0 7761 shows the logical equivalence to ax-ac 8101. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
Assertion
Ref Expression
ac2
Distinct variable group:   ,,,,,

Proof of Theorem ac2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-ac 8101 . 2
2 aceq0 7761 . 2
31, 2mpbir 200 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1530  wex 1531   wceq 1632   wcel 1696  wral 2556  wrex 2557  wreu 2558 This theorem is referenced by:  ac3  8104 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-ac 8101 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563
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