Theorem dcor 935

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 835	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orc 712	. . . . . 6 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	2	orcd 733	. . . . 5 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
4		df-dc 835	. . . . 5 ⊢ (DECID (𝜑 ∨ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
5	3, 4	sylibr 134	. . . 4 ⊢ (𝜑 → DECID (𝜑 ∨ 𝜓))
6	5	a1d 22	. . 3 ⊢ (𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
7		df-dc 835	. . . . 5 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
8		olc 711	. . . . . . . . 9 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
9	8	adantl 277	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜓))
10	9	orcd 733	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ 𝜓) → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
11	10, 4	sylibr 134	. . . . . 6 ⊢ ((¬ 𝜑 ∧ 𝜓) → DECID (𝜑 ∨ 𝜓))
12		ioran 752	. . . . . . . . 9 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
13	12	biimpri 133	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
14	13	olcd 734	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
15	14, 4	sylibr 134	. . . . . 6 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → DECID (𝜑 ∨ 𝜓))
16	11, 15	jaodan 797	. . . . 5 ⊢ ((¬ 𝜑 ∧ (𝜓 ∨ ¬ 𝜓)) → DECID (𝜑 ∨ 𝜓))
17	7, 16	sylan2b 287	. . . 4 ⊢ ((¬ 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∨ 𝜓))
18	17	ex 115	. . 3 ⊢ (¬ 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
19	6, 18	jaoi 716	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
20	1, 19	sylbi 121	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834

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 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-dc 835

This theorem is referenced by:  pm4.55dc  938  orandc  939  pm3.12dc  958  pm3.13dc  959  dn1dc  960  eueq3dc  2912  distrlem4prl  7583  distrlem4pru  7584  exfzdc  10240  lcmmndc  12062  isprm3  12118  lgsval  14408  lgsfvalg  14409  lgsfcl2  14410  lgsval2lem  14414  lgsdir2  14437  lgsne0  14442  lgsdirnn0  14451  lgsdinn0  14452