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 ) ) )