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Theorem decidin 10885
Description: If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)
Hypotheses
Ref Expression
decidin.ss (𝜑𝐴𝐵)
decidin.a (𝜑𝐴 DECIDin 𝐵)
decidin.b (𝜑𝐵 DECIDin 𝐶)
Assertion
Ref Expression
decidin (𝜑𝐴 DECIDin 𝐶)

Proof of Theorem decidin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 decidin.b . . . 4 (𝜑𝐵 DECIDin 𝐶)
2 decidi 10883 . . . 4 (𝐵 DECIDin 𝐶 → (𝑥𝐶 → (𝑥𝐵 ∨ ¬ 𝑥𝐵)))
31, 2syl 14 . . 3 (𝜑 → (𝑥𝐶 → (𝑥𝐵 ∨ ¬ 𝑥𝐵)))
4 decidin.a . . . . 5 (𝜑𝐴 DECIDin 𝐵)
5 decidi 10883 . . . . 5 (𝐴 DECIDin 𝐵 → (𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
64, 5syl 14 . . . 4 (𝜑 → (𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
7 decidin.ss . . . . . 6 (𝜑𝐴𝐵)
87ssneld 3010 . . . . 5 (𝜑 → (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
9 olc 665 . . . . 5 𝑥𝐴 → (𝑥𝐴 ∨ ¬ 𝑥𝐴))
108, 9syl6 33 . . . 4 (𝜑 → (¬ 𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
116, 10jaod 670 . . 3 (𝜑 → ((𝑥𝐵 ∨ ¬ 𝑥𝐵) → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
123, 11syld 44 . 2 (𝜑 → (𝑥𝐶 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
1312decidr 10884 1 (𝜑𝐴 DECIDin 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 662  wcel 1434  wss 2982   DECIDin wdcin 10881
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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-in 2988  df-ss 2995  df-dcin 10882
This theorem is referenced by:  sumdc2  10887
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