Theorem decidr 13174

Description: Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

Hypothesis

Ref	Expression
decidr.1	⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))

Assertion

Ref	Expression
decidr	⊢ (𝜑 → 𝐴 DECIDin 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem decidr

Step	Hyp	Ref	Expression
1		decidr.1	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
2		df-dc 821	. . . 4 ⊢ (DECID 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
3	1, 2	syl6ibr 161	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → DECID 𝑥 ∈ 𝐴))
4	3	alrimiv 1847	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐵 → DECID 𝑥 ∈ 𝐴))
5		df-dcin 13172	. . 3 ⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
6		df-ral 2422	. . 3 ⊢ (∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐵 → DECID 𝑥 ∈ 𝐴))
7	5, 6	bitri 183	. 2 ⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → DECID 𝑥 ∈ 𝐴))
8	4, 7	sylibr 133	1 ⊢ (𝜑 → 𝐴 DECIDin 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 820  ∀wal 1330   ∈ wcel 1481  ∀wral 2417   DECID_in wdcin 13171

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-5 1424  ax-gen 1426  ax-17 1507

This theorem depends on definitions:  df-bi 116  df-dc 821  df-ral 2422  df-dcin 13172

This theorem is referenced by:  decidin  13175  uzdcinzz  13176  sumdc2  13177