Theorem dcand 941

Description: A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) (Revised by BJ, 14-Nov-2024.)

Hypotheses

Ref	Expression
dcand.1	⊢ (𝜑 → DECID 𝜓)
dcand.2	⊢ (𝜑 → DECID 𝜒)

Assertion

Ref	Expression
dcand	⊢ (𝜑 → DECID (𝜓 ∧ 𝜒))

Proof of Theorem dcand

Step	Hyp	Ref	Expression
1		dcand.1	. . . 4 ⊢ (𝜑 → DECID 𝜓)
2		df-dc 843	. . . . 5 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
3		id 19	. . . . . . 7 ⊢ (¬ 𝜓 → ¬ 𝜓)
4	3	intnanrd 940	. . . . . 6 ⊢ (¬ 𝜓 → ¬ (𝜓 ∧ 𝜒))
5	4	orim2i 769	. . . . 5 ⊢ ((𝜓 ∨ ¬ 𝜓) → (𝜓 ∨ ¬ (𝜓 ∧ 𝜒)))
6	2, 5	sylbi 121	. . . 4 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ (𝜓 ∧ 𝜒)))
7	1, 6	syl 14	. . 3 ⊢ (𝜑 → (𝜓 ∨ ¬ (𝜓 ∧ 𝜒)))
8		dcand.2	. . . 4 ⊢ (𝜑 → DECID 𝜒)
9		df-dc 843	. . . . 5 ⊢ (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
10		id 19	. . . . . . 7 ⊢ (¬ 𝜒 → ¬ 𝜒)
11	10	intnand 939	. . . . . 6 ⊢ (¬ 𝜒 → ¬ (𝜓 ∧ 𝜒))
12	11	orim2i 769	. . . . 5 ⊢ ((𝜒 ∨ ¬ 𝜒) → (𝜒 ∨ ¬ (𝜓 ∧ 𝜒)))
13	9, 12	sylbi 121	. . . 4 ⊢ (DECID 𝜒 → (𝜒 ∨ ¬ (𝜓 ∧ 𝜒)))
14	8, 13	syl 14	. . 3 ⊢ (𝜑 → (𝜒 ∨ ¬ (𝜓 ∧ 𝜒)))
15		ordir 825	. . 3 ⊢ (((𝜓 ∧ 𝜒) ∨ ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜓 ∨ ¬ (𝜓 ∧ 𝜒)) ∧ (𝜒 ∨ ¬ (𝜓 ∧ 𝜒))))
16	7, 14, 15	sylanbrc 417	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∨ ¬ (𝜓 ∧ 𝜒)))
17		df-dc 843	. 2 ⊢ (DECID (𝜓 ∧ 𝜒) ↔ ((𝜓 ∧ 𝜒) ∨ ¬ (𝜓 ∧ 𝜒)))
18	16, 17	sylibr 134	1 ⊢ (𝜑 → DECID (𝜓 ∧ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842

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 717

This theorem depends on definitions:  df-bi 117  df-dc 843

This theorem is referenced by:  dcan  942  dcfi  7281  nn0n0n1ge2b  9678  infssfzcldc  10621  infssfzledc  10622  hashfibclem  11234  fzowrddc  11367  bitsinv1  12676  gcdsupex  12681  gcdsupcl  12682  gcdaddm  12708  nnwosdc  12763  lcmval  12788  lcmcllem  12792  lcmledvds  12795  prmdc  12855  pclemdc  13014  infpnlem2  13086  ballotfilemdifcfi  13172  ballotfilemiex  13191  nninfdclemcl  13286