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Theorem rabxmdc 3277
Description: Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
Assertion
Ref Expression
rabxmdc  |-  ( A. xDECID  ph 
->  A  =  ( { x  e.  A  |  ph }  u.  {
x  e.  A  |  -.  ph } ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rabxmdc
StepHypRef Expression
1 exmiddc 755 . . . . . 6  |-  (DECID  ph  ->  (
ph  \/  -.  ph )
)
21a1d 22 . . . . 5  |-  (DECID  ph  ->  ( x  e.  A  -> 
( ph  \/  -.  ph ) ) )
32alimi 1360 . . . 4  |-  ( A. xDECID  ph 
->  A. x ( x  e.  A  ->  ( ph  \/  -.  ph )
) )
4 df-ral 2328 . . . 4  |-  ( A. x  e.  A  ( ph  \/  -.  ph )  <->  A. x ( x  e.  A  ->  ( ph  \/  -.  ph ) ) )
53, 4sylibr 141 . . 3  |-  ( A. xDECID  ph 
->  A. x  e.  A  ( ph  \/  -.  ph ) )
6 rabid2 2503 . . 3  |-  ( A  =  { x  e.  A  |  ( ph  \/  -.  ph ) }  <->  A. x  e.  A  ( ph  \/  -.  ph ) )
75, 6sylibr 141 . 2  |-  ( A. xDECID  ph 
->  A  =  {
x  e.  A  | 
( ph  \/  -.  ph ) } )
8 unrab 3236 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } )  =  {
x  e.  A  | 
( ph  \/  -.  ph ) }
97, 8syl6eqr 2106 1  |-  ( A. xDECID  ph 
->  A  =  ( { x  e.  A  |  ph }  u.  {
x  e.  A  |  -.  ph } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 639  DECID wdc 753   A.wal 1257    = wceq 1259    e. wcel 1409   A.wral 2323   {crab 2327    u. cun 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-un 2950
This theorem is referenced by: (None)
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