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Theorem decidr 13033
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 820 . . . 4 (DECID 𝑥𝐴 ↔ (𝑥𝐴 ∨ ¬ 𝑥𝐴))
31, 2syl6ibr 161 . . 3 (𝜑 → (𝑥𝐵DECID 𝑥𝐴))
43alrimiv 1846 . 2 (𝜑 → ∀𝑥(𝑥𝐵DECID 𝑥𝐴))
5 df-dcin 13031 . . 3 (𝐴 DECIDin 𝐵 ↔ ∀𝑥𝐵 DECID 𝑥𝐴)
6 df-ral 2421 . . 3 (∀𝑥𝐵 DECID 𝑥𝐴 ↔ ∀𝑥(𝑥𝐵DECID 𝑥𝐴))
75, 6bitri 183 . 2 (𝐴 DECIDin 𝐵 ↔ ∀𝑥(𝑥𝐵DECID 𝑥𝐴))
84, 7sylibr 133 1 (𝜑𝐴 DECIDin 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 697  DECID wdc 819  wal 1329  wcel 1480  wral 2416   DECIDin wdcin 13030
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 13031
This theorem is referenced by:  decidin  13034  uzdcinzz  13035  sumdc2  13036
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