Metamath Proof Explorer


Theorem syl321anc

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

Ref Expression
Hypotheses syl3anc.1 φψ
syl3anc.2 φχ
syl3anc.3 φθ
syl3Xanc.4 φτ
syl23anc.5 φη
syl33anc.6 φζ
syl321anc.7 ψχθτηζσ
Assertion syl321anc φσ

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 syl321anc.7 ψχθτηζσ
8 4 5 jca φτη
9 1 2 3 8 6 7 syl311anc φσ