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Theorem decidi 13795
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 13794 . 2  |-  ( A DECIDin  B  <->  A. x  e.  B DECID  x  e.  A
)
2 df-dc 830 . . . 4  |-  (DECID  x  e.  A  <->  ( x  e.  A  \/  -.  x  e.  A ) )
32ralbii 2476 . . 3  |-  ( A. x  e.  B DECID  x  e.  A 
<-> 
A. x  e.  B  ( x  e.  A  \/  -.  x  e.  A
) )
4 eleq1 2233 . . . . 5  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54notbid 662 . . . . 5  |-  ( x  =  X  ->  ( -.  x  e.  A  <->  -.  X  e.  A ) )
64, 5orbi12d 788 . . . 4  |-  ( x  =  X  ->  (
( x  e.  A  \/  -.  x  e.  A
)  <->  ( X  e.  A  \/  -.  X  e.  A ) ) )
76rspccv 2831 . . 3  |-  ( A. x  e.  B  (
x  e.  A  \/  -.  x  e.  A
)  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
83, 7sylbi 120 . 2  |-  ( A. x  e.  B DECID  x  e.  A  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
91, 8sylbi 120 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 703  DECID wdc 829    = wceq 1348    e. wcel 2141   A.wral 2448   DECIDin wdcin 13793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dcin 13794
This theorem is referenced by:  decidin  13797
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