Metamath Proof Explorer


Theorem syl331anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)

Ref Expression
Hypotheses syl3anc.1 φψ
syl3anc.2 φχ
syl3anc.3 φθ
syl3Xanc.4 φτ
syl23anc.5 φη
syl33anc.6 φζ
syl133anc.7 φσ
syl331anc.8 ψχθτηζσρ
Assertion syl331anc φρ

Proof

Step Hyp Ref Expression
1 syl3anc.1 φψ
2 syl3anc.2 φχ
3 syl3anc.3 φθ
4 syl3Xanc.4 φτ
5 syl23anc.5 φη
6 syl33anc.6 φζ
7 syl133anc.7 φσ
8 syl331anc.8 ψχθτηζσρ
9 4 5 6 3jca φτηζ
10 1 2 3 9 7 8 syl311anc φρ