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 ( ( ( 𝜓𝜒𝜃 ) ∧ ( 𝜏𝜂 ) ∧ 𝜁 ) → 𝜎 )
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 ( ( ( 𝜓𝜒𝜃 ) ∧ ( 𝜏𝜂 ) ∧ 𝜁 ) → 𝜎 )
8 4 5 jca ( 𝜑 → ( 𝜏𝜂 ) )
9 1 2 3 8 6 7 syl311anc ( 𝜑𝜎 )