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Theorem ac2 8342
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 8343 is easier to understand.) Note: aceq0 8000 shows the logical equivalence to ax-ac 8340. (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 8340 . 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 8000 . 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 202 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 178    /\ wa 360   A.wal 1550   E.wex 1551   A.wral 2706   E.wrex 2707   E!wreu 2708
This theorem is referenced by:  ac3  8343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-ac 8340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-reu 2713
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