Metamath Proof Explorer


Theorem syl132anc

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 )
syl132anc.7 ( ( 𝜓 ∧ ( 𝜒𝜃𝜏 ) ∧ ( 𝜂𝜁 ) ) → 𝜎 )
Assertion syl132anc ( 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 syl132anc.7 ( ( 𝜓 ∧ ( 𝜒𝜃𝜏 ) ∧ ( 𝜂𝜁 ) ) → 𝜎 )
8 5 6 jca ( 𝜑 → ( 𝜂𝜁 ) )
9 1 2 3 4 8 7 syl131anc ( 𝜑𝜎 )