Metamath Proof Explorer


Theorem syl213anc

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