Theorem syl3an1

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995)

	Ref	Expression
Hypotheses	syl3an1.1	⊢ ( 𝜑 → 𝜓 )
	syl3an1.2	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
Assertion	syl3an1	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syl3an1.1	⊢ ( 𝜑 → 𝜓 )
2		syl3an1.2	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
3	1	3anim1i	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
4	3 2	syl	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )