Theorem decidi 15865

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 15864	. 2 ⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
2		df-dc 837	. . . 4 ⊢ (DECID 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
3	2	ralbii 2513	. . 3 ⊢ (∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
4		eleq1 2269	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
5	4	notbid 669	. . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐴 ↔ ¬ 𝑋 ∈ 𝐴))
6	4, 5	orbi12d 795	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴) ↔ (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
7	6	rspccv 2878	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴) → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
8	3, 7	sylbi 121	. 2 ⊢ (∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
9	1, 8	sylbi 121	1 ⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 710  DECID wdc 836   = wceq 1373   ∈ wcel 2177  ∀wral 2485   DECID_in wdcin 15863

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-dcin 15864

This theorem is referenced by:  decidin  15867