Theorem decidi 13002

Description: Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

Assertion

Ref	Expression
decidi	⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))

Proof of Theorem decidi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dcin 13001	. 2 ⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
2		df-dc 820	. . . 4 ⊢ (DECID 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
3	2	ralbii 2441	. . 3 ⊢ (∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
4		eleq1 2202	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
5	4	notbid 656	. . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐴 ↔ ¬ 𝑋 ∈ 𝐴))
6	4, 5	orbi12d 782	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴) ↔ (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
7	6	rspccv 2786	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴) → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
8	3, 7	sylbi 120	. 2 ⊢ (∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
9	1, 8	sylbi 120	1 ⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480  ∀wral 2416   DECID_in wdcin 13000

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dcin 13001

This theorem is referenced by:  decidin  13004