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Theorem dcor 854
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 754 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orc 643 . . . . . 6 (𝜑 → (𝜑𝜓))
32orcd 662 . . . . 5 (𝜑 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
4 df-dc 754 . . . . 5 (DECID (𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
53, 4sylibr 141 . . . 4 (𝜑DECID (𝜑𝜓))
65a1d 22 . . 3 (𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
7 df-dc 754 . . . . 5 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
8 olc 642 . . . . . . . . 9 (𝜓 → (𝜑𝜓))
98adantl 266 . . . . . . . 8 ((¬ 𝜑𝜓) → (𝜑𝜓))
109orcd 662 . . . . . . 7 ((¬ 𝜑𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
1110, 4sylibr 141 . . . . . 6 ((¬ 𝜑𝜓) → DECID (𝜑𝜓))
12 ioran 679 . . . . . . . . 9 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
1312biimpri 128 . . . . . . . 8 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
1413olcd 663 . . . . . . 7 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
1514, 4sylibr 141 . . . . . 6 ((¬ 𝜑 ∧ ¬ 𝜓) → DECID (𝜑𝜓))
1611, 15jaodan 721 . . . . 5 ((¬ 𝜑 ∧ (𝜓 ∨ ¬ 𝜓)) → DECID (𝜑𝜓))
177, 16sylan2b 275 . . . 4 ((¬ 𝜑DECID 𝜓) → DECID (𝜑𝜓))
1817ex 112 . . 3 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
196, 18jaoi 646 . 2 ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓DECID (𝜑𝜓)))
201, 19sylbi 118 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  pm4.55dc  857  pm3.12dc  876  pm3.13dc  877  dn1dc  878  eueq3dc  2738  distrlem4prl  6740  distrlem4pru  6741
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