Theorem dcor 943

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 842	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orc 719	. . . . . 6 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	2	orcd 740	. . . . 5 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
4		df-dc 842	. . . . 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 842	. . . . 5 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
8		olc 718	. . . . . . . . 9 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
9	8	adantl 277	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜓))
10	9	orcd 740	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ 𝜓) → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
11	10, 4	sylibr 134	. . . . . 6 ⊢ ((¬ 𝜑 ∧ 𝜓) → DECID (𝜑 ∨ 𝜓))
12		ioran 759	. . . . . . . . 9 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
13	12	biimpri 133	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
14	13	olcd 741	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
15	14, 4	sylibr 134	. . . . . 6 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → DECID (𝜑 ∨ 𝜓))
16	11, 15	jaodan 804	. . . . 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 723	. 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 715  DECID wdc 841

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

This theorem depends on definitions:  df-bi 117  df-dc 842

This theorem is referenced by:  pm4.55dc  946  orandc  947  pm3.12dc  966  pm3.13dc  967  dn1dc  968  eueq3dc  2980  distrlem4prl  7804  distrlem4pru  7805  exfzdc  10487  lcmmndc  12636  isprm3  12692  perfectlem2  15727  lgsval  15736  lgsfvalg  15737  lgsfcl2  15738  lgsval2lem  15742  lgsdir2  15765  lgsne0  15770  lgsdirnn0  15779  lgsdinn0  15780  2lgs  15836  2lgsoddprm  15845  eupth2lem3lem4fi  16327  eupth2lem3lem7fi  16328  cndcap  16684