Metamath Proof Explorer
Description: A double syllogism inference. (Contributed by SN, 15-Sep-2024)
|
|
Ref |
Expression |
|
Hypotheses |
syl3an12.1 |
⊢ ( 𝜑 → 𝜓 ) |
|
|
syl3an12.2 |
⊢ ( 𝜒 → 𝜃 ) |
|
|
syl3an12.s |
⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) → 𝜂 ) |
|
Assertion |
syl3an12 |
⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
syl3an12.1 |
⊢ ( 𝜑 → 𝜓 ) |
2 |
|
syl3an12.2 |
⊢ ( 𝜒 → 𝜃 ) |
3 |
|
syl3an12.s |
⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) → 𝜂 ) |
4 |
|
id |
⊢ ( 𝜏 → 𝜏 ) |
5 |
1 2 4 3
|
syl3an |
⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 ) |