Metamath Proof Explorer


Theorem syl131anc

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 )
syl131anc.6 ( ( 𝜓 ∧ ( 𝜒𝜃𝜏 ) ∧ 𝜂 ) → 𝜁 )
Assertion syl131anc ( 𝜑𝜁 )

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 syl131anc.6 ( ( 𝜓 ∧ ( 𝜒𝜃𝜏 ) ∧ 𝜂 ) → 𝜁 )
7 2 3 4 3jca ( 𝜑 → ( 𝜒𝜃𝜏 ) )
8 1 7 5 6 syl3anc ( 𝜑𝜁 )