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Mirrors > Home > ILE Home > Th. List > pm5.62dc | GIF version |
Description: Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.) |
Ref | Expression |
---|---|
pm5.62dc | ⊢ (DECID 𝜓 → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 821 | . 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓)) | |
2 | ordir 807 | . . . 4 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ ((𝜑 ∨ ¬ 𝜓) ∧ (𝜓 ∨ ¬ 𝜓))) | |
3 | 2 | simplbi 272 | . . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓)) |
4 | 2 | simplbi2 383 | . . . 4 ⊢ ((𝜑 ∨ ¬ 𝜓) → ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∧ 𝜓) ∨ ¬ 𝜓))) |
5 | 4 | com12 30 | . . 3 ⊢ ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → ((𝜑 ∧ 𝜓) ∨ ¬ 𝜓))) |
6 | 3, 5 | impbid2 142 | . 2 ⊢ ((𝜓 ∨ ¬ 𝜓) → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))) |
7 | 1, 6 | sylbi 120 | 1 ⊢ (DECID 𝜓 → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 820 |
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 699 |
This theorem depends on definitions: df-bi 116 df-dc 821 |
This theorem is referenced by: (None) |
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