Theorem syld3an1

Description: A syllogism inference. (Contributed by NM, 7-Jul-2008) (Proof shortened by Wolf Lammen, 26-Jun-2022)

	Ref	Expression
Hypotheses	syld3an1.1	⊢ ( ( 𝜒 ∧ 𝜓 ∧ 𝜃 ) → 𝜑 )
	syld3an1.2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion	syld3an1	⊢ ( ( 𝜒 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syld3an1.1	⊢ ( ( 𝜒 ∧ 𝜓 ∧ 𝜃 ) → 𝜑 )
2		syld3an1.2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
3		simp2	⊢ ( ( 𝜒 ∧ 𝜓 ∧ 𝜃 ) → 𝜓 )
4		simp3	⊢ ( ( 𝜒 ∧ 𝜓 ∧ 𝜃 ) → 𝜃 )
5	1 3 4 2	syl3anc	⊢ ( ( 𝜒 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )