Theorem decidi 16417

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 16416	. 2 ⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
2		df-dc 842	. . . 4 ⊢ (DECID 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
3	2	ralbii 2538	. . 3 ⊢ (∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
4		eleq1 2294	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
5	4	notbid 673	. . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐴 ↔ ¬ 𝑋 ∈ 𝐴))
6	4, 5	orbi12d 800	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴) ↔ (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
7	6	rspccv 2907	. . 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 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510   DECID_in wdcin 16415

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dcin 16416

This theorem is referenced by:  decidin  16419