Theorem dcand 940

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 842	. . . . 5 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
3		id 19	. . . . . . 7 ⊢ (¬ 𝜓 → ¬ 𝜓)
4	3	intnanrd 939	. . . . . 6 ⊢ (¬ 𝜓 → ¬ (𝜓 ∧ 𝜒))
5	4	orim2i 768	. . . . 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 842	. . . . 5 ⊢ (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
10		id 19	. . . . . . 7 ⊢ (¬ 𝜒 → ¬ 𝜒)
11	10	intnand 938	. . . . . 6 ⊢ (¬ 𝜒 → ¬ (𝜓 ∧ 𝜒))
12	11	orim2i 768	. . . . 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 824	. . 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 842	. 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 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:  dcan  941  dcfi  7185  nn0n0n1ge2b  9564  fzowrddc  11237  bitsinv1  12546  gcdsupex  12551  gcdsupcl  12552  gcdaddm  12578  nnwosdc  12633  lcmval  12658  lcmcllem  12662  lcmledvds  12665  prmdc  12725  pclemdc  12884  infpnlem2  12956  nninfdclemcl  13092