Theorem sylan9bbr

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

	Ref	Expression
Hypotheses	sylan9bbr.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	sylan9bbr.2	$$⊢\theta \to \left(\chi \leftrightarrow \tau \right)$$
Assertion	sylan9bbr	$$⊢\left(\theta \wedge \phi \right)\to \left(\psi \leftrightarrow \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		sylan9bbr.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		sylan9bbr.2	$$⊢\theta \to \left(\chi \leftrightarrow \tau \right)$$
3	1 2	sylan9bb	$$⊢\left(\phi \wedge \theta \right)\to \left(\psi \leftrightarrow \tau \right)$$
4	3	ancoms	$$⊢\left(\theta \wedge \phi \right)\to \left(\psi \leftrightarrow \tau \right)$$