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Theorem decidr 14408
Description: Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
Hypothesis
Ref Expression
decidr.1 (𝜑 → (𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
Assertion
Ref Expression
decidr (𝜑𝐴 DECIDin 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem decidr
StepHypRef Expression
1 decidr.1 . . . 4 (𝜑 → (𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
2 df-dc 835 . . . 4 (DECID 𝑥𝐴 ↔ (𝑥𝐴 ∨ ¬ 𝑥𝐴))
31, 2syl6ibr 162 . . 3 (𝜑 → (𝑥𝐵DECID 𝑥𝐴))
43alrimiv 1874 . 2 (𝜑 → ∀𝑥(𝑥𝐵DECID 𝑥𝐴))
5 df-dcin 14406 . . 3 (𝐴 DECIDin 𝐵 ↔ ∀𝑥𝐵 DECID 𝑥𝐴)
6 df-ral 2460 . . 3 (∀𝑥𝐵 DECID 𝑥𝐴 ↔ ∀𝑥(𝑥𝐵DECID 𝑥𝐴))
75, 6bitri 184 . 2 (𝐴 DECIDin 𝐵 ↔ ∀𝑥(𝑥𝐵DECID 𝑥𝐴))
84, 7sylibr 134 1 (𝜑𝐴 DECIDin 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 708  DECID wdc 834  wal 1351  wcel 2148  wral 2455   DECIDin wdcin 14405
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-5 1447  ax-gen 1449  ax-17 1526
This theorem depends on definitions:  df-bi 117  df-dc 835  df-ral 2460  df-dcin 14406
This theorem is referenced by:  decidin  14409  uzdcinzz  14410  sumdc2  14411
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