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Theorem decidin 11343
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 11341 . . . 4 (𝐵 DECIDin 𝐶 → (𝑥𝐶 → (𝑥𝐵 ∨ ¬ 𝑥𝐵)))
31, 2syl 14 . . 3 (𝜑 → (𝑥𝐶 → (𝑥𝐵 ∨ ¬ 𝑥𝐵)))
4 decidin.a . . . . 5 (𝜑𝐴 DECIDin 𝐵)
5 decidi 11341 . . . . 5 (𝐴 DECIDin 𝐵 → (𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
64, 5syl 14 . . . 4 (𝜑 → (𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
7 decidin.ss . . . . . 6 (𝜑𝐴𝐵)
87ssneld 3025 . . . . 5 (𝜑 → (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
9 olc 667 . . . . 5 𝑥𝐴 → (𝑥𝐴 ∨ ¬ 𝑥𝐴))
108, 9syl6 33 . . . 4 (𝜑 → (¬ 𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
116, 10jaod 672 . . 3 (𝜑 → ((𝑥𝐵 ∨ ¬ 𝑥𝐵) → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
123, 11syld 44 . 2 (𝜑 → (𝑥𝐶 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))
1312decidr 11342 1 (𝜑𝐴 DECIDin 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 664  wcel 1438  wss 2997   DECIDin wdcin 11339
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3003  df-ss 3010  df-dcin 11340
This theorem is referenced by:  sumdc2  11345
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