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Theorem pm2.13dc 855
Description: A decidable proposition or its triple negation is true. Theorem *2.13 of [WhiteheadRussell] p. 101 with decidability condition added. (Contributed by Jim Kingdon, 13-May-2018.)
Assertion
Ref Expression
pm2.13dc (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑))

Proof of Theorem pm2.13dc
StepHypRef Expression
1 df-dc 805 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 notnotrdc 813 . . . . 5 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
32con3d 605 . . . 4 (DECID 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
43orim2d 762 . . 3 (DECID 𝜑 → ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ¬ ¬ ¬ 𝜑)))
51, 4syl5bi 151 . 2 (DECID 𝜑 → (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑)))
65pm2.43i 49 1 (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 682  DECID wdc 804
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 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-dc 805
This theorem is referenced by: (None)
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