Theorem dcan 935

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

Assertion

Ref	Expression
dcan	⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ 𝜓))

Proof of Theorem dcan

Step	Hyp	Ref	Expression
1		simpl 109	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID 𝜑)
2		simpr 110	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID 𝜓)
3	1, 2	dcand 934	1 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by:  dcan2  936  dcbi  938  annimdc  939  pm4.55dc  940  orandc  941  anordc  958  xordidc  1410  gcdmndc  12081