Metamath Proof Explorer


Theorem syl322anc

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
syl322anc.8 ψ χ θ τ η ζ σ ρ
Assertion syl322anc 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 syl322anc.8 ψ χ θ τ η ζ σ ρ
9 6 7 jca φ ζ σ
10 1 2 3 4 5 9 8 syl321anc φ ρ