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Theorem decidi 15287
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 15286 . 2  |-  ( A DECIDin  B  <->  A. x  e.  B DECID  x  e.  A
)
2 df-dc 836 . . . 4  |-  (DECID  x  e.  A  <->  ( x  e.  A  \/  -.  x  e.  A ) )
32ralbii 2500 . . 3  |-  ( A. x  e.  B DECID  x  e.  A 
<-> 
A. x  e.  B  ( x  e.  A  \/  -.  x  e.  A
) )
4 eleq1 2256 . . . . 5  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54notbid 668 . . . . 5  |-  ( x  =  X  ->  ( -.  x  e.  A  <->  -.  X  e.  A ) )
64, 5orbi12d 794 . . . 4  |-  ( x  =  X  ->  (
( x  e.  A  \/  -.  x  e.  A
)  <->  ( X  e.  A  \/  -.  X  e.  A ) ) )
76rspccv 2861 . . 3  |-  ( A. x  e.  B  (
x  e.  A  \/  -.  x  e.  A
)  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
83, 7sylbi 121 . 2  |-  ( A. x  e.  B DECID  x  e.  A  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
91, 8sylbi 121 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 709  DECID wdc 835    = wceq 1364    e. wcel 2164   A.wral 2472   DECIDin wdcin 15285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-dcin 15286
This theorem is referenced by:  decidin  15289
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