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Theorem pm2.13dc 813
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 777 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 notnotrdc 785 . . . . 5 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
32con3d 594 . . . 4 (DECID 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
43orim2d 735 . . 3 (DECID 𝜑 → ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ¬ ¬ ¬ 𝜑)))
51, 4syl5bi 150 . 2 (DECID 𝜑 → (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑)))
65pm2.43i 48 1 (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 662  DECID wdc 776
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 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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