Metamath Proof Explorer


Theorem syl333anc

Description: A syllogism inference combined with contraction. (Contributed by NM, 10-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
syl333anc.9 φ μ
syl333anc.10 ψ χ θ τ η ζ σ ρ μ λ
Assertion syl333anc φ λ

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 syl333anc.9 φ μ
10 syl333anc.10 ψ χ θ τ η ζ σ ρ μ λ
11 7 8 9 3jca φ σ ρ μ
12 1 2 3 4 5 6 11 10 syl331anc φ λ