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Theorem decidr 13056
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 820 . . . 4  |-  (DECID  x  e.  A  <->  ( x  e.  A  \/  -.  x  e.  A ) )
31, 2syl6ibr 161 . . 3  |-  ( ph  ->  ( x  e.  B  -> DECID  x  e.  A ) )
43alrimiv 1846 . 2  |-  ( ph  ->  A. x ( x  e.  B  -> DECID  x  e.  A
) )
5 df-dcin 13054 . . 3  |-  ( A DECIDin  B  <->  A. x  e.  B DECID  x  e.  A
)
6 df-ral 2421 . . 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 697  DECID wdc 819   A.wal 1329    e. wcel 1480   A.wral 2416   DECIDin wdcin 13053
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 1423  ax-gen 1425  ax-17 1506
This theorem depends on definitions:  df-bi 116  df-dc 820  df-ral 2421  df-dcin 13054
This theorem is referenced by:  decidin  13057  uzdcinzz  13058  sumdc2  13059
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