Theorem syl3an9b

Description: Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995)

	Ref	Expression
Hypotheses	syl3an9b.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	syl3an9b.2	⊢ ( 𝜃 → ( 𝜒 ↔ 𝜏 ) )
	syl3an9b.3	⊢ ( 𝜂 → ( 𝜏 ↔ 𝜁 ) )
Assertion	syl3an9b	⊢ ( ( 𝜑 ∧ 𝜃 ∧ 𝜂 ) → ( 𝜓 ↔ 𝜁 ) )

Proof

Step	Hyp	Ref	Expression
1		syl3an9b.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		syl3an9b.2	⊢ ( 𝜃 → ( 𝜒 ↔ 𝜏 ) )
3		syl3an9b.3	⊢ ( 𝜂 → ( 𝜏 ↔ 𝜁 ) )
4	1 2	sylan9bb	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 ↔ 𝜏 ) )
5	4 3	sylan9bb	⊢ ( ( ( 𝜑 ∧ 𝜃 ) ∧ 𝜂 ) → ( 𝜓 ↔ 𝜁 ) )
6	5	3impa	⊢ ( ( 𝜑 ∧ 𝜃 ∧ 𝜂 ) → ( 𝜓 ↔ 𝜁 ) )