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Theorem axac3 8308
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8307 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 8307 . . 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 1552 . 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 8010 . 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 201 1  |- CHOICE
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1546   E.wex 1547  CHOICEwac 7960
This theorem is referenced by:  ackm  8309  axac  8311  axaci  8312  cardeqv  8313  fin71ac  8375  lbsex  16200  ptcls  17609  ptcmp  18050  axac10  27002
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-ac2 8307
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ac 7961
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