Metamath Proof Explorer


Theorem syl23anc

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 )
syl23anc.6 ( ( ( 𝜓𝜒 ) ∧ ( 𝜃𝜏𝜂 ) ) → 𝜁 )
Assertion syl23anc ( 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 syl23anc.6 ( ( ( 𝜓𝜒 ) ∧ ( 𝜃𝜏𝜂 ) ) → 𝜁 )
7 1 2 jca ( 𝜑 → ( 𝜓𝜒 ) )
8 7 3 4 5 6 syl13anc ( 𝜑𝜁 )