Theorem dcor 843

Description: A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)

Assertion

Ref	Expression
dcor	⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))

Proof of Theorem dcor

Step	Hyp	Ref	Expression
1		df-dc 743	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orc 633	. . . . . 6 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	2	orcd 652	. . . . 5 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
4		df-dc 743	. . . . 5 ⊢ (DECID (𝜑 ∨ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
5	3, 4	sylibr 137	. . . 4 ⊢ (𝜑 → DECID (𝜑 ∨ 𝜓))
6	5	a1d 22	. . 3 ⊢ (𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
7		df-dc 743	. . . . 5 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
8		olc 632	. . . . . . . . 9 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
9	8	adantl 262	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜓))
10	9	orcd 652	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ 𝜓) → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
11	10, 4	sylibr 137	. . . . . 6 ⊢ ((¬ 𝜑 ∧ 𝜓) → DECID (𝜑 ∨ 𝜓))
12		ioran 669	. . . . . . . . 9 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
13	12	biimpri 124	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
14	13	olcd 653	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
15	14, 4	sylibr 137	. . . . . 6 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → DECID (𝜑 ∨ 𝜓))
16	11, 15	jaodan 710	. . . . 5 ⊢ ((¬ 𝜑 ∧ (𝜓 ∨ ¬ 𝜓)) → DECID (𝜑 ∨ 𝜓))
17	7, 16	sylan2b 271	. . . 4 ⊢ ((¬ 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∨ 𝜓))
18	17	ex 108	. . 3 ⊢ (¬ 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
19	6, 18	jaoi 636	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
20	1, 19	sylbi 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm4.55dc  846  pm3.12dc  865  pm3.13dc  866  dn1dc  867  eueq3dc  2715  distrlem4prl  6682  distrlem4pru  6683