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 ) |
|
|
syl213anc.7 |
⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ∧ ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) ) → 𝜎 ) |
|
Assertion |
syl213anc |
⊢ ( 𝜑 → 𝜎 ) |
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 |
|
syl213anc.7 |
⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ∧ ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) ) → 𝜎 ) |
8 |
1 2
|
jca |
⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) |
9 |
8 3 4 5 6 7
|
syl113anc |
⊢ ( 𝜑 → 𝜎 ) |