Metamath Proof Explorer
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)
|
|
Ref |
Expression |
|
Hypotheses |
syl3anc.1 |
⊢ ( 𝜑 → 𝜓 ) |
|
|
syl3anc.2 |
⊢ ( 𝜑 → 𝜒 ) |
|
|
syl3anc.3 |
⊢ ( 𝜑 → 𝜃 ) |
|
|
syl3Xanc.4 |
⊢ ( 𝜑 → 𝜏 ) |
|
|
syl23anc.5 |
⊢ ( 𝜑 → 𝜂 ) |
|
|
syl33anc.6 |
⊢ ( 𝜑 → 𝜁 ) |
|
|
syl133anc.7 |
⊢ ( 𝜑 → 𝜎 ) |
|
|
syl331anc.8 |
⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) ∧ 𝜎 ) → 𝜌 ) |
|
Assertion |
syl331anc |
⊢ ( 𝜑 → 𝜌 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
syl3anc.1 |
⊢ ( 𝜑 → 𝜓 ) |
2 |
|
syl3anc.2 |
⊢ ( 𝜑 → 𝜒 ) |
3 |
|
syl3anc.3 |
⊢ ( 𝜑 → 𝜃 ) |
4 |
|
syl3Xanc.4 |
⊢ ( 𝜑 → 𝜏 ) |
5 |
|
syl23anc.5 |
⊢ ( 𝜑 → 𝜂 ) |
6 |
|
syl33anc.6 |
⊢ ( 𝜑 → 𝜁 ) |
7 |
|
syl133anc.7 |
⊢ ( 𝜑 → 𝜎 ) |
8 |
|
syl331anc.8 |
⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) ∧ 𝜎 ) → 𝜌 ) |
9 |
4 5 6
|
3jca |
⊢ ( 𝜑 → ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) ) |
10 |
1 2 3 9 7 8
|
syl311anc |
⊢ ( 𝜑 → 𝜌 ) |