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Metamath Proof Explorer

Theorem sylan9r

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993)

Ref Expression
Hypotheses sylan9r.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
sylan9r.2 ⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) )
Assertion sylan9r ( ( 𝜃 ∧ 𝜑 ) → ( 𝜓 → 𝜏 ) )

Proof

Step Hyp Ref Expression
1 sylan9r.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 sylan9r.2 ⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) )
3 1 2 syl9r ⊢ ( 𝜃 → ( 𝜑 → ( 𝜓 → 𝜏 ) ) )
4 3 imp ⊢ ( ( 𝜃 ∧ 𝜑 ) → ( 𝜓 → 𝜏 ) )