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Theorem axac3 8349
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8348 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
Assertion
Ref Expression
axac3  |- CHOICE

Proof of Theorem axac3
Dummy variables  w  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-ac2 8348 . . 3  |-  E. y A. z E. w A. v ( ( y  e.  x  /\  (
z  e.  y  -> 
( ( w  e.  x  /\  -.  y  =  w )  /\  z  e.  w ) ) )  \/  ( -.  y  e.  x  /\  (
z  e.  x  -> 
( ( w  e.  z  /\  w  e.  y )  /\  (
( v  e.  z  /\  v  e.  y )  ->  v  =  w ) ) ) ) )
21ax-gen 1556 . 2  |-  A. x E. y A. z E. w A. v ( ( y  e.  x  /\  ( z  e.  y  ->  ( ( w  e.  x  /\  -.  y  =  w )  /\  z  e.  w
) ) )  \/  ( -.  y  e.  x  /\  ( z  e.  x  ->  (
( w  e.  z  /\  w  e.  y )  /\  ( ( v  e.  z  /\  v  e.  y )  ->  v  =  w ) ) ) ) )
3 dfackm 8051 . 2  |-  (CHOICE  <->  A. x E. y A. z E. w A. v ( ( y  e.  x  /\  ( z  e.  y  ->  ( ( w  e.  x  /\  -.  y  =  w )  /\  z  e.  w
) ) )  \/  ( -.  y  e.  x  /\  ( z  e.  x  ->  (
( w  e.  z  /\  w  e.  y )  /\  ( ( v  e.  z  /\  v  e.  y )  ->  v  =  w ) ) ) ) ) )
42, 3mpbir 202 1  |- CHOICE
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360   A.wal 1550   E.wex 1551  CHOICEwac 8001
This theorem is referenced by:  ackm  8350  axac  8352  axaci  8353  cardeqv  8354  fin71ac  8416  lbsex  16242  ptcls  17653  ptcmp  18094  axac10  27118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-ac2 8348
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ac 8002
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