Metamath Proof Explorer


Theorem syl322anc

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
syl322anc.8 ψχθτηζσρ
Assertion syl322anc 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 syl322anc.8 ψχθτηζσρ
9 6 7 jca φζσ
10 1 2 3 4 5 9 8 syl321anc φρ