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Theorem ac2 8297
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 8298 is easier to understand.) Note: aceq0 7955 shows the logical equivalence to ax-ac 8295. (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.
StepHypRef Expression
1 ax-ac 8295 . 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 7955 . 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
) ) )
31, 2mpbir 201 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
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547   A.wral 2666   E.wrex 2667   E!wreu 2668
This theorem is referenced by:  ac3  8298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-ac 8295
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-reu 2673
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