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Metamath Proof Explorer

Theorem syl32anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)

Ref Expression
Hypotheses syl3anc.1 ⊢ ( 𝜑 → 𝜓 )
syl3anc.2 ⊢ ( 𝜑 → 𝜒 )
syl3anc.3 ⊢ ( 𝜑 → 𝜃 )
syl3Xanc.4 ⊢ ( 𝜑 → 𝜏 )
syl23anc.5 ⊢ ( 𝜑 → 𝜂 )
syl32anc.6 ⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ) → 𝜁 )
Assertion syl32anc ( 𝜑 → 𝜁 )

Proof

Step Hyp Ref Expression
1 syl3anc.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3anc.2 ⊢ ( 𝜑 → 𝜒 )
3 syl3anc.3 ⊢ ( 𝜑 → 𝜃 )
4 syl3Xanc.4 ⊢ ( 𝜑 → 𝜏 )
5 syl23anc.5 ⊢ ( 𝜑 → 𝜂 )
6 syl32anc.6 ⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ) → 𝜁 )
7 4 5 jca ⊢ ( 𝜑 → ( 𝜏 ∧ 𝜂 ) )
8 1 2 3 7 6 syl31anc ⊢ ( 𝜑 → 𝜁 )