Theorem ac2 4756

Description: Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single conjunction. (If you want to figure it out, the rewritten equivalent ac3 4757 is easier to understand.) Note: aceq0 4740 shows the logical equivalence to ax-ac 4754.

Assertion

Ref	Expression
ac2	|- E.yA.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

Step	Hyp	Ref	Expression
1		ax-ac 4754	. 2 |- E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))
2		aceq0 4740	. 2 |- (E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u) <-> E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v)))
3	1, 2	mpbir 190	1 |- E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  A.wral 1648  E.wrex 1649  E!wreu 1650

This theorem is referenced by:  ac3 4757  ac7 4758

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-11o 1220  ax-ext 1462  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-reu 1654

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