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Mirrors > Home > ILE Home > Th. List > pm2.13dc | GIF version |
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.) |
Ref | Expression |
---|---|
pm2.13dc | ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 743 | . . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | notnotrdc 751 | . . . . 5 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑)) | |
3 | 2 | con3d 561 | . . . 4 ⊢ (DECID 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑)) |
4 | 3 | orim2d 702 | . . 3 ⊢ (DECID 𝜑 → ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ¬ ¬ ¬ 𝜑))) |
5 | 1, 4 | syl5bi 141 | . 2 ⊢ (DECID 𝜑 → (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑))) |
6 | 5 | pm2.43i 43 | 1 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 629 DECID wdc 742 |
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 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: (None) |
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