Theorem bi2bian9

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

	Ref	Expression
Hypotheses	bi2an9.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	bi2an9.2	$$⊢\theta \to \left(\tau \leftrightarrow \eta \right)$$
Assertion	bi2bian9	$$⊢\left(\phi \wedge \theta \right)\to \left(\left(\psi \leftrightarrow \tau \right)\leftrightarrow \left(\chi \leftrightarrow \eta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bi2an9.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		bi2an9.2	$$⊢\theta \to \left(\tau \leftrightarrow \eta \right)$$
3	1	adantr	$$⊢\left(\phi \wedge \theta \right)\to \left(\psi \leftrightarrow \chi \right)$$
4	2	adantl	$$⊢\left(\phi \wedge \theta \right)\to \left(\tau \leftrightarrow \eta \right)$$
5	3 4	bibi12d	$$⊢\left(\phi \wedge \theta \right)\to \left(\left(\psi \leftrightarrow \tau \right)\leftrightarrow \left(\chi \leftrightarrow \eta \right)\right)$$