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Theorem decidr 11053
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 779 . . . 4  |-  (DECID  x  e.  A  <->  ( x  e.  A  \/  -.  x  e.  A ) )
31, 2syl6ibr 160 . . 3  |-  ( ph  ->  ( x  e.  B  -> DECID  x  e.  A ) )
43alrimiv 1799 . 2  |-  ( ph  ->  A. x ( x  e.  B  -> DECID  x  e.  A
) )
5 df-dcin 11051 . . 3  |-  ( A DECIDin  B  <->  A. x  e.  B DECID  x  e.  A
)
6 df-ral 2360 . . 3  |-  ( A. x  e.  B DECID  x  e.  A 
<-> 
A. x ( x  e.  B  -> DECID  x  e.  A
) )
75, 6bitri 182 . 2  |-  ( A DECIDin  B  <->  A. x
( x  e.  B  -> DECID  x  e.  A ) )
84, 7sylibr 132 1  |-  ( ph  ->  A DECIDin  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 662  DECID wdc 778   A.wal 1285    e. wcel 1436   A.wral 2355   DECIDin wdcin 11050
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-5 1379  ax-gen 1381  ax-17 1462
This theorem depends on definitions:  df-bi 115  df-dc 779  df-ral 2360  df-dcin 11051
This theorem is referenced by:  decidin  11054  uzdcinzz  11055  sumdc2  11056
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