Theorem syl2ani

Description: A syllogism inference. (Contributed by NM, 3-Aug-1999)

	Ref	Expression
Hypotheses	syl2ani.1	⊢ ( 𝜑 → 𝜒 )
	syl2ani.2	⊢ ( 𝜂 → 𝜃 )
	syl2ani.3	⊢ ( 𝜓 → ( ( 𝜒 ∧ 𝜃 ) → 𝜏 ) )
Assertion	syl2ani	⊢ ( 𝜓 → ( ( 𝜑 ∧ 𝜂 ) → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		syl2ani.1	⊢ ( 𝜑 → 𝜒 )
2		syl2ani.2	⊢ ( 𝜂 → 𝜃 )
3		syl2ani.3	⊢ ( 𝜓 → ( ( 𝜒 ∧ 𝜃 ) → 𝜏 ) )
4	2 3	sylan2i	⊢ ( 𝜓 → ( ( 𝜒 ∧ 𝜂 ) → 𝜏 ) )
5	1 4	sylani	⊢ ( 𝜓 → ( ( 𝜑 ∧ 𝜂 ) → 𝜏 ) )