Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  decidi GIF version

Theorem decidi 11052
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 11051 . 2 (𝐴 DECIDin 𝐵 ↔ ∀𝑥𝐵 DECID 𝑥𝐴)
2 df-dc 779 . . . 4 (DECID 𝑥𝐴 ↔ (𝑥𝐴 ∨ ¬ 𝑥𝐴))
32ralbii 2380 . . 3 (∀𝑥𝐵 DECID 𝑥𝐴 ↔ ∀𝑥𝐵 (𝑥𝐴 ∨ ¬ 𝑥𝐴))
4 eleq1 2147 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
54notbid 625 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥𝐴 ↔ ¬ 𝑋𝐴))
64, 5orbi12d 740 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴 ∨ ¬ 𝑥𝐴) ↔ (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
76rspccv 2711 . . 3 (∀𝑥𝐵 (𝑥𝐴 ∨ ¬ 𝑥𝐴) → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
83, 7sylbi 119 . 2 (∀𝑥𝐵 DECID 𝑥𝐴 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
91, 8sylbi 119 1 (𝐴 DECIDin 𝐵 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 662  DECID wdc 778   = wceq 1287  wcel 1436  wral 2355   DECIDin wdcin 11050
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-dc 779  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-dcin 11051
This theorem is referenced by:  decidin  11054
  Copyright terms: Public domain W3C validator