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Metamath Proof Explorer
Description: A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993) (Proof shortened by Josh Purinton, 29-Dec-2000)
|
|
Ref |
Expression |
|
Hypotheses |
syl9.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
|
|
syl9.2 |
⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) ) |
|
Assertion |
syl9 |
⊢ ( 𝜑 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
syl9.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
| 2 |
|
syl9.2 |
⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) ) |
| 3 |
2
|
a1i |
⊢ ( 𝜑 → ( 𝜃 → ( 𝜒 → 𝜏 ) ) ) |
| 4 |
1 3
|
syl5d |
⊢ ( 𝜑 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) ) |