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Mirrors > Home > ILE Home > Th. List > syldd | GIF version |
Description: Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.) |
Ref | Expression |
---|---|
syldd.1 | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
syldd.2 | ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) |
Ref | Expression |
---|---|
syldd | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syldd.2 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) | |
2 | syldd.1 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | |
3 | imim2 55 | . 2 ⊢ ((𝜃 → 𝜏) → ((𝜒 → 𝜃) → (𝜒 → 𝜏))) | |
4 | 1, 2, 3 | syl6c 66 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: syl5d 68 syl6d 70 syl10 1422 ordiso2 6992 oddprmdvds 12273 |
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