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Mirrors > Home > ILE Home > Th. List > pm2.82 | GIF version |
Description: Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
pm2.82 | ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑 ∨ 𝜓) ∨ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 5 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ ¬ 𝜒) → (𝜑 ∨ 𝜓))) | |
2 | pm2.24 551 | . . . 4 ⊢ (𝜒 → (¬ 𝜒 → 𝜓)) | |
3 | 2 | orim2d 702 | . . 3 ⊢ (𝜒 → ((𝜑 ∨ ¬ 𝜒) → (𝜑 ∨ 𝜓))) |
4 | 1, 3 | jaoi 636 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → ((𝜑 ∨ ¬ 𝜒) → (𝜑 ∨ 𝜓))) |
5 | 4 | orim1d 701 | 1 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑 ∨ 𝜓) ∨ 𝜃))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 629 |
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 |
This theorem is referenced by: (None) |
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