Theorem ac2 8862

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 8863 is easier to understand.) Note: aceq0 8520 shows the logical equivalence to ax-ac 8860. (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.

Step	Hyp	Ref	Expression
1		ax-ac 8860	. 2
2		aceq0 8520	. 2
3	1, 2	mpbir 209	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  A.wral 2807  E.wrex 2808  E!wreu 2809

This theorem is referenced by:  ac3  8863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-ac 8860

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-reu 2814