Theorem ac2 4670

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 4671 is easier to understand.) Note: aceq0 4654 shows the logical equivalence to ax-ac 4668.

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 4668	. 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 4654	. 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 950  E.wex 956   = wceq 1099   e. wcel 1105  A.wral 1621  E.wrex 1622  E!wreu 1623

This theorem is referenced by:  ac3 4671  ac7 4672

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-11o 1202  ax-ext 1436  ax-ac 4668

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-cleq 1446  df-clel 1449  df-ral 1625  df-rex 1626  df-reu 1627

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