Theorem sylani

Description: A syllogism inference. (Contributed by NM, 2-May-1996)

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

Proof

Step	Hyp	Ref	Expression
1		sylani.1	⊢ ( 𝜑 → 𝜒 )
2		sylani.2	⊢ ( 𝜓 → ( ( 𝜒 ∧ 𝜃 ) → 𝜏 ) )
3	1	a1i	⊢ ( 𝜓 → ( 𝜑 → 𝜒 ) )
4	3 2	syland	⊢ ( 𝜓 → ( ( 𝜑 ∧ 𝜃 ) → 𝜏 ) )