Theorem bi2bian9

Description: Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004)

	Ref	Expression
Hypotheses	bi2an9.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	bi2an9.2	⊢ ( 𝜃 → ( 𝜏 ↔ 𝜂 ) )
Assertion	bi2bian9	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ↔ 𝜏 ) ↔ ( 𝜒 ↔ 𝜂 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bi2an9.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bi2an9.2	⊢ ( 𝜃 → ( 𝜏 ↔ 𝜂 ) )
3	1	adantr	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 ↔ 𝜒 ) )
4	2	adantl	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜏 ↔ 𝜂 ) )
5	3 4	bibi12d	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ↔ 𝜏 ) ↔ ( 𝜒 ↔ 𝜂 ) ) )