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Theorem dcim 809
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 803 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 df-dc 803 . . . . . . . 8 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
32anbi2i 450 . . . . . . 7 ((𝜑DECID 𝜓) ↔ (𝜑 ∧ (𝜓 ∨ ¬ 𝜓)))
4 andi 790 . . . . . . 7 ((𝜑 ∧ (𝜓 ∨ ¬ 𝜓)) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
53, 4bitri 183 . . . . . 6 ((𝜑DECID 𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
6 pm3.4 329 . . . . . . 7 ((𝜑𝜓) → (𝜑𝜓))
7 annimim 658 . . . . . . 7 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
86, 7orim12i 731 . . . . . 6 (((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
95, 8sylbi 120 . . . . 5 ((𝜑DECID 𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
10 df-dc 803 . . . . 5 (DECID (𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
119, 10sylibr 133 . . . 4 ((𝜑DECID 𝜓) → DECID (𝜑𝜓))
1211ex 114 . . 3 (𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
13 ax-in2 587 . . . . 5 𝜑 → (𝜑𝜓))
1413a1d 22 . . . 4 𝜑 → (DECID 𝜓 → (𝜑𝜓)))
15 orc 684 . . . . 5 ((𝜑𝜓) → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
1615, 10sylibr 133 . . . 4 ((𝜑𝜓) → DECID (𝜑𝜓))
1714, 16syl6 33 . . 3 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
1812, 17jaoi 688 . 2 ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓DECID (𝜑𝜓)))
191, 18sylbi 120 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 680  DECID wdc 802
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 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-dc 803
This theorem is referenced by:  pm4.79dc  871  pm5.11dc  877  dcbi  903  annimdc  904
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