Theorem pm2.13dc 886

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

Step	Hyp	Ref	Expression
1		df-dc 836	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		notnotrdc 844	. . . . 5 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3	2	con3d 632	. . . 4 ⊢ (DECID 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
4	3	orim2d 789	. . 3 ⊢ (DECID 𝜑 → ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ¬ ¬ ¬ 𝜑)))
5	1, 4	biimtrid 152	. 2 ⊢ (DECID 𝜑 → (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑)))
6	5	pm2.43i 49	1 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 709  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by: (None)