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Mirrors > Home > ILE Home > Th. List > pm5.63dc | GIF version |
Description: Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.) |
Ref | Expression |
---|---|
pm5.63dc | ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 830 | . . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | ordi 811 | . . . 4 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜓)) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜑 ∨ 𝜓))) | |
3 | 2 | simplbi2 383 | . . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ (¬ 𝜑 ∧ 𝜓)))) |
4 | 1, 3 | sylbi 120 | . 2 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) → (𝜑 ∨ (¬ 𝜑 ∧ 𝜓)))) |
5 | 2 | simprbi 273 | . 2 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜓)) → (𝜑 ∨ 𝜓)) |
6 | 4, 5 | impbid1 141 | 1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ 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: (None) |
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