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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 776 | . 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓)) | |
2 | ordir 763 | . . . 4 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ ((𝜑 ∨ ¬ 𝜓) ∧ (𝜓 ∨ ¬ 𝜓))) | |
3 | 2 | simplbi 268 | . . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓)) |
4 | 2 | simplbi2 377 | . . . 4 ⊢ ((𝜑 ∨ ¬ 𝜓) → ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∧ 𝜓) ∨ ¬ 𝜓))) |
5 | 4 | com12 30 | . . 3 ⊢ ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → ((𝜑 ∧ 𝜓) ∨ ¬ 𝜓))) |
6 | 3, 5 | impbid2 141 | . 2 ⊢ ((𝜓 ∨ ¬ 𝜓) → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))) |
7 | 1, 6 | sylbi 119 | 1 ⊢ (DECID 𝜓 → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 DECID wdc 775 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-dc 776 |
This theorem is referenced by: (None) |
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