Theorem syl3anb

Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005)

	Ref	Expression
Hypotheses	syl3anb.1	⊢ ( 𝜑 ↔ 𝜓 )
	syl3anb.2	⊢ ( 𝜒 ↔ 𝜃 )
	syl3anb.3	⊢ ( 𝜏 ↔ 𝜂 )
	syl3anb.4	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
Assertion	syl3anb	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )

Proof

Step	Hyp	Ref	Expression
1		syl3anb.1	⊢ ( 𝜑 ↔ 𝜓 )
2		syl3anb.2	⊢ ( 𝜒 ↔ 𝜃 )
3		syl3anb.3	⊢ ( 𝜏 ↔ 𝜂 )
4		syl3anb.4	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
5	1 2 3	3anbi123i	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) ↔ ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) )
6	5 4	sylbi	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )