Theorem sylan9bb

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995)

	Ref	Expression
Hypotheses	sylan9bb.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	sylan9bb.2	⊢ ( 𝜃 → ( 𝜒 ↔ 𝜏 ) )
Assertion	sylan9bb	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 ↔ 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		sylan9bb.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		sylan9bb.2	⊢ ( 𝜃 → ( 𝜒 ↔ 𝜏 ) )
3	1	adantr	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 ↔ 𝜒 ) )
4	2	adantl	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜒 ↔ 𝜏 ) )
5	3 4	bitrd	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 ↔ 𝜏 ) )