Theorem syld3an2

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

	Ref	Expression
Hypotheses	syld3an2.1	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜓 )
	syld3an2.2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion	syld3an2	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syld3an2.1	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜓 )
2		syld3an2.2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
3		simp1	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜑 )
4		simp3	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜃 )
5	3 1 4 2	syl3anc	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )