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Mirrors > Home > ILE Home > Th. List > pm2.25dc | GIF version |
Description: Elimination of disjunction based on a disjunction, for a decidable proposition. Based on theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
pm2.25dc | ⊢ (DECID 𝜑 → (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 743 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | orel1 644 | . . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓)) | |
3 | 2 | orim2i 678 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓))) |
4 | 1, 3 | sylbi 114 | 1 ⊢ (DECID 𝜑 → (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓))) |
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-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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