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Theorem axac3 8794
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8793 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 8793 . . 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 1637 . 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 8496 . 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 366    /\ wa 367   A.wal 1401   E.wex 1631  CHOICEwac 8446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-ac2 8793
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ac 8447
This theorem is referenced by:  ackm  8795  axac  8797  axaci  8798  cardeqv  8799  fin71ac  8861  lbsex  18021  ptcls  20299  ptcmp  20740  axac10  35301
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