This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem syl323anc

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 )
syl233anc.8 ⊢ ( 𝜑 → 𝜌 )
syl323anc.9 ⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ∧ ( 𝜁 ∧ 𝜎 ∧ 𝜌 ) ) → 𝜇 )
Assertion syl323anc ( 𝜑 → 𝜇 )

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 syl233anc.8 ⊢ ( 𝜑 → 𝜌 )
9 syl323anc.9 ⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ∧ ( 𝜁 ∧ 𝜎 ∧ 𝜌 ) ) → 𝜇 )
10 4 5 jca ⊢ ( 𝜑 → ( 𝜏 ∧ 𝜂 ) )
11 1 2 3 10 6 7 8 9 syl313anc ⊢ ( 𝜑 → 𝜇 )