Theorem dcand 938

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 840	. . . . 5 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
3		id 19	. . . . . . 7 ⊢ (¬ 𝜓 → ¬ 𝜓)
4	3	intnanrd 937	. . . . . 6 ⊢ (¬ 𝜓 → ¬ (𝜓 ∧ 𝜒))
5	4	orim2i 766	. . . . 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 840	. . . . 5 ⊢ (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
10		id 19	. . . . . . 7 ⊢ (¬ 𝜒 → ¬ 𝜒)
11	10	intnand 936	. . . . . 6 ⊢ (¬ 𝜒 → ¬ (𝜓 ∧ 𝜒))
12	11	orim2i 766	. . . . 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 822	. . 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 840	. 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 713  DECID wdc 839

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840

This theorem is referenced by:  dcan  939  dcfi  7174  nn0n0n1ge2b  9552  fzowrddc  11221  bitsinv1  12516  gcdsupex  12521  gcdsupcl  12522  gcdaddm  12548  nnwosdc  12603  lcmval  12628  lcmcllem  12632  lcmledvds  12635  prmdc  12695  pclemdc  12854  infpnlem2  12926  nninfdclemcl  13062