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Theorem decidi 13212
 Description: Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
Assertion
Ref Expression
decidi (𝐴 DECIDin 𝐵 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))

Proof of Theorem decidi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-dcin 13211 . 2 (𝐴 DECIDin 𝐵 ↔ ∀𝑥𝐵 DECID 𝑥𝐴)
2 df-dc 821 . . . 4 (DECID 𝑥𝐴 ↔ (𝑥𝐴 ∨ ¬ 𝑥𝐴))
32ralbii 2446 . . 3 (∀𝑥𝐵 DECID 𝑥𝐴 ↔ ∀𝑥𝐵 (𝑥𝐴 ∨ ¬ 𝑥𝐴))
4 eleq1 2204 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
54notbid 657 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥𝐴 ↔ ¬ 𝑋𝐴))
64, 5orbi12d 783 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴 ∨ ¬ 𝑥𝐴) ↔ (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
76rspccv 2792 . . 3 (∀𝑥𝐵 (𝑥𝐴 ∨ ¬ 𝑥𝐴) → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
83, 7sylbi 120 . 2 (∀𝑥𝐵 DECID 𝑥𝐴 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
91, 8sylbi 120 1 (𝐴 DECIDin 𝐵 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 820   = wceq 1332   ∈ wcel 2112  ∀wral 2418   DECIDin wdcin 13210 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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-v 2693  df-dcin 13211 This theorem is referenced by:  decidin  13214
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