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Mirrors > Home > ILE Home > Th. List > pm2.1dc | GIF version |
Description: Commuted law of the excluded middle for a decidable proposition. Based on theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by Jim Kingdon, 25-Mar-2018.) |
Ref | Expression |
---|---|
pm2.1dc | ⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 830 | . . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | orcom 723 | . . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑)) | |
3 | 1, 2 | bitri 183 | . 2 ⊢ (DECID 𝜑 ↔ (¬ 𝜑 ∨ 𝜑)) |
4 | 3 | biimpi 119 | 1 ⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 703 DECID wdc 829 |
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-io 704 |
This theorem depends on definitions: df-bi 116 df-dc 830 |
This theorem is referenced by: pm2.6dc 857 rabrsndc 3649 |
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