Metamath Proof Explorer


Theorem syl213anc

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 )
syl213anc.7 ( ( ( 𝜓𝜒 ) ∧ 𝜃 ∧ ( 𝜏𝜂𝜁 ) ) → 𝜎 )
Assertion syl213anc ( 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 syl213anc.7 ( ( ( 𝜓𝜒 ) ∧ 𝜃 ∧ ( 𝜏𝜂𝜁 ) ) → 𝜎 )
8 1 2 jca ( 𝜑 → ( 𝜓𝜒 ) )
9 8 3 4 5 6 7 syl113anc ( 𝜑𝜎 )