Theorem syl3an

Description: A triple syllogism inference. (Contributed by NM, 13-May-2004)

	Ref	Expression
Hypotheses	syl3an.1	⊢ ( 𝜑 → 𝜓 )
	syl3an.2	⊢ ( 𝜒 → 𝜃 )
	syl3an.3	⊢ ( 𝜏 → 𝜂 )
	syl3an.4	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
Assertion	syl3an	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )

Proof

Step	Hyp	Ref	Expression
1		syl3an.1	⊢ ( 𝜑 → 𝜓 )
2		syl3an.2	⊢ ( 𝜒 → 𝜃 )
3		syl3an.3	⊢ ( 𝜏 → 𝜂 )
4		syl3an.4	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
5	1 2 3	3anim123i	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) )
6	5 4	syl	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )