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Metamath Proof Explorer

Theorem syl232anc

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 )
syl33anc.6 ⊢ ( 𝜑 → 𝜁 )
syl133anc.7 ⊢ ( 𝜑 → 𝜎 )
syl232anc.8 ⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ∧ 𝜂 ) ∧ ( 𝜁 ∧ 𝜎 ) ) → 𝜌 )
Assertion syl232anc ( 𝜑 → 𝜌 )

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 syl33anc.6 ⊢ ( 𝜑 → 𝜁 )
7 syl133anc.7 ⊢ ( 𝜑 → 𝜎 )
8 syl232anc.8 ⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ∧ 𝜂 ) ∧ ( 𝜁 ∧ 𝜎 ) ) → 𝜌 )
9 6 7 jca ⊢ ( 𝜑 → ( 𝜁 ∧ 𝜎 ) )
10 1 2 3 4 5 9 8 syl231anc ⊢ ( 𝜑 → 𝜌 )