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Theorem decidi 14169
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 14168 . 2 (𝐴 DECIDin 𝐵 ↔ ∀𝑥𝐵 DECID 𝑥𝐴)
2 df-dc 835 . . . 4 (DECID 𝑥𝐴 ↔ (𝑥𝐴 ∨ ¬ 𝑥𝐴))
32ralbii 2483 . . 3 (∀𝑥𝐵 DECID 𝑥𝐴 ↔ ∀𝑥𝐵 (𝑥𝐴 ∨ ¬ 𝑥𝐴))
4 eleq1 2240 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
54notbid 667 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥𝐴 ↔ ¬ 𝑋𝐴))
64, 5orbi12d 793 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴 ∨ ¬ 𝑥𝐴) ↔ (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
76rspccv 2838 . . 3 (∀𝑥𝐵 (𝑥𝐴 ∨ ¬ 𝑥𝐴) → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
83, 7sylbi 121 . 2 (∀𝑥𝐵 DECID 𝑥𝐴 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
91, 8sylbi 121 1 (𝐴 DECIDin 𝐵 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455   DECIDin wdcin 14167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dcin 14168
This theorem is referenced by:  decidin  14171
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