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Metamath Proof Explorer

Theorem syl9r

Description: A nested syllogism inference with different antecedents. (Contributed by NM, 14-May-1993)

Ref Expression
Hypotheses syl9r.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
syl9r.2 ⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) )
Assertion syl9r ( 𝜃 → ( 𝜑 → ( 𝜓 → 𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 syl9r.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 syl9r.2 ⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) )
3 1 2 syl9 ⊢ ( 𝜑 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) )
4 3 com12 ⊢ ( 𝜃 → ( 𝜑 → ( 𝜓 → 𝜏 ) ) )