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Theorem dcor 919
Description: A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
Assertion
Ref Expression
dcor (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))

Proof of Theorem dcor
StepHypRef Expression
1 df-dc 820 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orc 701 . . . . . 6 (𝜑 → (𝜑𝜓))
32orcd 722 . . . . 5 (𝜑 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
4 df-dc 820 . . . . 5 (DECID (𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
53, 4sylibr 133 . . . 4 (𝜑DECID (𝜑𝜓))
65a1d 22 . . 3 (𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
7 df-dc 820 . . . . 5 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
8 olc 700 . . . . . . . . 9 (𝜓 → (𝜑𝜓))
98adantl 275 . . . . . . . 8 ((¬ 𝜑𝜓) → (𝜑𝜓))
109orcd 722 . . . . . . 7 ((¬ 𝜑𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
1110, 4sylibr 133 . . . . . 6 ((¬ 𝜑𝜓) → DECID (𝜑𝜓))
12 ioran 741 . . . . . . . . 9 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
1312biimpri 132 . . . . . . . 8 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
1413olcd 723 . . . . . . 7 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
1514, 4sylibr 133 . . . . . 6 ((¬ 𝜑 ∧ ¬ 𝜓) → DECID (𝜑𝜓))
1611, 15jaodan 786 . . . . 5 ((¬ 𝜑 ∧ (𝜓 ∨ ¬ 𝜓)) → DECID (𝜑𝜓))
177, 16sylan2b 285 . . . 4 ((¬ 𝜑DECID 𝜓) → DECID (𝜑𝜓))
1817ex 114 . . 3 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
196, 18jaoi 705 . 2 ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓DECID (𝜑𝜓)))
201, 19sylbi 120 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697  DECID wdc 819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-dc 820
This theorem is referenced by:  pm4.55dc  922  orandc  923  pm3.12dc  942  pm3.13dc  943  dn1dc  944  eueq3dc  2858  distrlem4prl  7392  distrlem4pru  7393  exfzdc  10017  lcmmndc  11743  isprm3  11799
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