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Theorem decidr 12930
Description: Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
Hypothesis
Ref Expression
decidr.1  |-  ( ph  ->  ( x  e.  B  ->  ( x  e.  A  \/  -.  x  e.  A
) ) )
Assertion
Ref Expression
decidr  |-  ( ph  ->  A DECIDin  B )
Distinct variable groups:    x, A    x, B    ph, x

Proof of Theorem decidr
StepHypRef Expression
1 decidr.1 . . . 4  |-  ( ph  ->  ( x  e.  B  ->  ( x  e.  A  \/  -.  x  e.  A
) ) )
2 df-dc 805 . . . 4  |-  (DECID  x  e.  A  <->  ( x  e.  A  \/  -.  x  e.  A ) )
31, 2syl6ibr 161 . . 3  |-  ( ph  ->  ( x  e.  B  -> DECID  x  e.  A ) )
43alrimiv 1830 . 2  |-  ( ph  ->  A. x ( x  e.  B  -> DECID  x  e.  A
) )
5 df-dcin 12928 . . 3  |-  ( A DECIDin  B  <->  A. x  e.  B DECID  x  e.  A
)
6 df-ral 2398 . . 3  |-  ( A. x  e.  B DECID  x  e.  A 
<-> 
A. x ( x  e.  B  -> DECID  x  e.  A
) )
75, 6bitri 183 . 2  |-  ( A DECIDin  B  <->  A. x
( x  e.  B  -> DECID  x  e.  A ) )
84, 7sylibr 133 1  |-  ( ph  ->  A DECIDin  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 682  DECID wdc 804   A.wal 1314    e. wcel 1465   A.wral 2393   DECIDin wdcin 12927
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-5 1408  ax-gen 1410  ax-17 1491
This theorem depends on definitions:  df-bi 116  df-dc 805  df-ral 2398  df-dcin 12928
This theorem is referenced by:  decidin  12931  uzdcinzz  12932  sumdc2  12933
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