Theorem dcan 942

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 941	1 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  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:  dcan2  943  dcbi  945  annimdc  946  pm4.55dc  947  orandc  948  anordc  965  xordidc  1444  gcdmndc  12587