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Theorem axac3 8835
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8834 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 8834 . . 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 1596 . 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 8537 . 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 209 1  |- CHOICE
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1372   E.wex 1591  CHOICEwac 8487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-ac2 8834
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ac 8488
This theorem is referenced by:  ackm  8836  axac  8838  axaci  8839  cardeqv  8840  fin71ac  8902  lbsex  17589  ptcls  19847  ptcmp  20288  axac10  30570
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