Theorem syld3an3

Description: A syllogism inference. (Contributed by NM, 20-May-2007)

	Ref	Expression
Hypotheses	syld3an3.1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
	syld3an3.2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion	syld3an3	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syld3an3.1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
2		syld3an3.2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
3		simp1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜑 )
4		simp2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜓 )
5	3 4 1 2	syl3anc	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜏 )