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Theorem dcan 876
Description: A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
Assertion
Ref Expression
dcan (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))

Proof of Theorem dcan
StepHypRef Expression
1 simpl 107 . . . . . 6 ((¬ 𝜑𝜓) → ¬ 𝜑)
21intnanrd 875 . . . . 5 ((¬ 𝜑𝜓) → ¬ (𝜑𝜓))
32orim2i 711 . . . 4 (((𝜑𝜓) ∨ (¬ 𝜑𝜓)) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
4 simpr 108 . . . . . 6 (((𝜑 ∨ ¬ 𝜑) ∧ ¬ 𝜓) → ¬ 𝜓)
54intnand 874 . . . . 5 (((𝜑 ∨ ¬ 𝜑) ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
65olcd 686 . . . 4 (((𝜑 ∨ ¬ 𝜑) ∧ ¬ 𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
73, 6jaoi 669 . . 3 ((((𝜑𝜓) ∨ (¬ 𝜑𝜓)) ∨ ((𝜑 ∨ ¬ 𝜑) ∧ ¬ 𝜓)) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
8 df-dc 777 . . . . 5 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
9 df-dc 777 . . . . 5 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
108, 9anbi12i 448 . . . 4 ((DECID 𝜑DECID 𝜓) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜓 ∨ ¬ 𝜓)))
11 andi 765 . . . 4 (((𝜑 ∨ ¬ 𝜑) ∧ (𝜓 ∨ ¬ 𝜓)) ↔ (((𝜑 ∨ ¬ 𝜑) ∧ 𝜓) ∨ ((𝜑 ∨ ¬ 𝜑) ∧ ¬ 𝜓)))
12 andir 766 . . . . 5 (((𝜑 ∨ ¬ 𝜑) ∧ 𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜓)))
1312orbi1i 713 . . . 4 ((((𝜑 ∨ ¬ 𝜑) ∧ 𝜓) ∨ ((𝜑 ∨ ¬ 𝜑) ∧ ¬ 𝜓)) ↔ (((𝜑𝜓) ∨ (¬ 𝜑𝜓)) ∨ ((𝜑 ∨ ¬ 𝜑) ∧ ¬ 𝜓)))
1410, 11, 133bitri 204 . . 3 ((DECID 𝜑DECID 𝜓) ↔ (((𝜑𝜓) ∨ (¬ 𝜑𝜓)) ∨ ((𝜑 ∨ ¬ 𝜑) ∧ ¬ 𝜓)))
15 df-dc 777 . . 3 (DECID (𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
167, 14, 153imtr4i 199 . 2 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
1716ex 113 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  dcbi  878  annimdc  879  pm4.55dc  880  orandc  881  anordc  898  xordidc  1331  nn0n0n1ge2b  8508  gcdmndc  10484  gcdsupex  10493  gcdsupcl  10494  gcdaddm  10519  lcmval  10589  lcmcllem  10593  lcmledvds  10596
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