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Theorem pm2.13dc 778
 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 742 . . 3 (DECID φ ↔ (φ ¬ φ))
2 notnot2dc 750 . . . . 5 (DECID φ → (¬ ¬ φφ))
32con3d 560 . . . 4 (DECID φ → (¬ φ → ¬ ¬ ¬ φ))
43orim2d 701 . . 3 (DECID φ → ((φ ¬ φ) → (φ ¬ ¬ ¬ φ)))
51, 4syl5bi 141 . 2 (DECID φ → (DECID φ → (φ ¬ ¬ ¬ φ)))
65pm2.43i 43 1 (DECID φ → (φ ¬ ¬ ¬ φ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 628  DECID wdc 741 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629 This theorem depends on definitions:  df-bi 110  df-dc 742 This theorem is referenced by: (None)
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