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Theorem decidr 12413
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 784 . . . 4 (DECID 𝑥𝐴 ↔ (𝑥𝐴 ∨ ¬ 𝑥𝐴))
31, 2syl6ibr 161 . . 3 (𝜑 → (𝑥𝐵DECID 𝑥𝐴))
43alrimiv 1809 . 2 (𝜑 → ∀𝑥(𝑥𝐵DECID 𝑥𝐴))
5 df-dcin 12411 . . 3 (𝐴 DECIDin 𝐵 ↔ ∀𝑥𝐵 DECID 𝑥𝐴)
6 df-ral 2375 . . 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 667  DECID wdc 783  wal 1294  wcel 1445  wral 2370   DECIDin wdcin 12410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-17 1471
This theorem depends on definitions:  df-bi 116  df-dc 784  df-ral 2375  df-dcin 12411
This theorem is referenced by:  decidin  12414  uzdcinzz  12415  sumdc2  12416
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