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Theorem dcim 822
Description: An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
Assertion
Ref Expression
dcim (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))

Proof of Theorem dcim
StepHypRef Expression
1 df-dc 781 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 df-dc 781 . . . . . . . 8 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
32anbi2i 445 . . . . . . 7 ((𝜑DECID 𝜓) ↔ (𝜑 ∧ (𝜓 ∨ ¬ 𝜓)))
4 andi 767 . . . . . . 7 ((𝜑 ∧ (𝜓 ∨ ¬ 𝜓)) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
53, 4bitri 182 . . . . . 6 ((𝜑DECID 𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
6 pm3.4 326 . . . . . . 7 ((𝜑𝜓) → (𝜑𝜓))
7 annimim 820 . . . . . . 7 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
86, 7orim12i 711 . . . . . 6 (((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
95, 8sylbi 119 . . . . 5 ((𝜑DECID 𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
10 df-dc 781 . . . . 5 (DECID (𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
119, 10sylibr 132 . . . 4 ((𝜑DECID 𝜓) → DECID (𝜑𝜓))
1211ex 113 . . 3 (𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
13 ax-in2 580 . . . . 5 𝜑 → (𝜑𝜓))
1413a1d 22 . . . 4 𝜑 → (DECID 𝜓 → (𝜑𝜓)))
15 orc 668 . . . . 5 ((𝜑𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
1615, 10sylibr 132 . . . 4 ((𝜑𝜓) → DECID (𝜑𝜓))
1714, 16syl6 33 . . 3 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
1812, 17jaoi 671 . 2 ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓DECID (𝜑𝜓)))
191, 18sylbi 119 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 664  DECID wdc 780
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  pm4.79dc  847  pm5.11dc  853  dcbi  882  annimdc  883
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