Metamath Proof Explorer


Theorem syl32anc

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
syl32anc.6 ψ χ θ τ η ζ
Assertion syl32anc φ ζ

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 syl32anc.6 ψ χ θ τ η ζ
7 4 5 jca φ τ η
8 1 2 3 7 6 syl31anc φ ζ