Theorem dcor 948

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 847	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orc 724	. . . . . 6 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	2	orcd 745	. . . . 5 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
4		df-dc 847	. . . . 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 847	. . . . 5 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
8		olc 723	. . . . . . . . 9 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
9	8	adantl 277	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜓))
10	9	orcd 745	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ 𝜓) → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
11	10, 4	sylibr 134	. . . . . 6 ⊢ ((¬ 𝜑 ∧ 𝜓) → DECID (𝜑 ∨ 𝜓))
12		ioran 764	. . . . . . . . 9 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
13	12	biimpri 133	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
14	13	olcd 746	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
15	14, 4	sylibr 134	. . . . . 6 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → DECID (𝜑 ∨ 𝜓))
16	11, 15	jaodan 809	. . . . 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 728	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
20	1, 19	sylbi 121	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-dc 847

This theorem is used by:  pm4.55dc  951  orandc  952  pm3.12dc  971  pm3.13dc  972  dn1dc  973  eueq3dc  3000  distrlem4prl  7951  distrlem4pru  7952  exfzdc  10661  lcmmndc  12842  isprm3  12898  perfectlem2  16120  lgsval  16135  lgsfvalg  16136  lgsfcl2  16137  lgsval2lem  16141  lgsdir2  16164  lgsne0  16169  lgsdirnn0  16178  lgsdinn0  16179  2lgs  16235  2lgsoddprm  16244  eupth2lem3lem4fi  16726  eupth2lem3lem7fi  16727  cndcap  17121