Theorem syl3an3

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995) (Proof shortened by Wolf Lammen, 26-Jun-2022)

	Ref	Expression
Hypotheses	syl3an3.1	⊢ ( 𝜑 → 𝜃 )
	syl3an3.2	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
Assertion	syl3an3	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜑 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syl3an3.1	⊢ ( 𝜑 → 𝜃 )
2		syl3an3.2	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
3	1	3anim3i	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜑 ) → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
4	3 2	syl	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜑 ) → 𝜏 )