Theorem ac2 9283

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 9284 is easier to understand.) Note: aceq0 8941 shows the logical equivalence to ax-ac 9281. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

Assertion

Ref	Expression
ac2	|- E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u )

Distinct variable group:   x, y, z, w, v, u

Proof of Theorem ac2

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-ac 9281	. 2 |- E. y A. z A. w ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) )
2		aceq0 8941	. 2 |- ( E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u ) <-> E. y A. z A. w ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) ) )
3	1, 2	mpbir 221	1 |- E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   A.wral 2912   E.wrex 2913   E!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-ac 9281

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919

This theorem is referenced by:  ac3  9284