Theorem sylan9bb

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

	Ref	Expression
Hypotheses	sylan9bb.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	sylan9bb.2	$$⊢\theta \to \left(\chi \leftrightarrow \tau \right)$$
Assertion	sylan9bb	$$⊢\left(\phi \wedge \theta \right)\to \left(\psi \leftrightarrow \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		sylan9bb.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		sylan9bb.2	$$⊢\theta \to \left(\chi \leftrightarrow \tau \right)$$
3	1	adantr	$$⊢\left(\phi \wedge \theta \right)\to \left(\psi \leftrightarrow \chi \right)$$
4	2	adantl	$$⊢\left(\phi \wedge \theta \right)\to \left(\chi \leftrightarrow \tau \right)$$
5	3 4	bitrd	$$⊢\left(\phi \wedge \theta \right)\to \left(\psi \leftrightarrow \tau \right)$$