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Theorem decidi 11037
Description: Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
Assertion
Ref Expression
decidi  |-  ( A DECIDin  B  -> 
( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A
) ) )

Proof of Theorem decidi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-dcin 11036 . 2  |-  ( A DECIDin  B  <->  A. x  e.  B DECID  x  e.  A
)
2 df-dc 777 . . . 4  |-  (DECID  x  e.  A  <->  ( x  e.  A  \/  -.  x  e.  A ) )
32ralbii 2378 . . 3  |-  ( A. x  e.  B DECID  x  e.  A 
<-> 
A. x  e.  B  ( x  e.  A  \/  -.  x  e.  A
) )
4 eleq1 2145 . . . . 5  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54notbid 625 . . . . 5  |-  ( x  =  X  ->  ( -.  x  e.  A  <->  -.  X  e.  A ) )
64, 5orbi12d 740 . . . 4  |-  ( x  =  X  ->  (
( x  e.  A  \/  -.  x  e.  A
)  <->  ( X  e.  A  \/  -.  X  e.  A ) ) )
76rspccv 2709 . . 3  |-  ( A. x  e.  B  (
x  e.  A  \/  -.  x  e.  A
)  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
83, 7sylbi 119 . 2  |-  ( A. x  e.  B DECID  x  e.  A  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
91, 8sylbi 119 1  |-  ( A DECIDin  B  -> 
( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434   A.wral 2353   DECIDin wdcin 11035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-dcin 11036
This theorem is referenced by:  decidin  11039
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