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| Mirrors > Home > MPE Home > Th. List > syldd | Structured version Visualization version GIF version | ||
| Description: Nested syllogism deduction. Deduction associated with syld 47. Double deduction associated with syl 17. (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 58 | . 2 ⊢ ((𝜃 → 𝜏) → ((𝜒 → 𝜃) → (𝜒 → 𝜏))) | |
| 4 | 1, 2, 3 | syl6c 70 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) |
| Colors of variables: wff setvar 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 73 syl6d 75 syl10 79 fvf1pr 7327 tfinds 7881 soseq 8184 tz7.49 8485 php3 9249 dffi2 9463 ordiso2 9555 rankuni2b 9893 oddprmdvds 16941 brbtwn2 28920 bj-exalims 36635 prtlem60 38854 lvoli2 39583 |
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