Theorem 3anbi23d

Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006)

	Ref	Expression
Hypotheses	3anbi12d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	3anbi12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
Assertion	3anbi23d	$$⊢\phi \to \left(\left(\eta \wedge \psi \wedge \theta \right)\leftrightarrow \left(\eta \wedge \chi \wedge \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3anbi12d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		3anbi12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
3		biidd	$$⊢\phi \to \left(\eta \leftrightarrow \eta \right)$$
4	3 1 2	3anbi123d	$$⊢\phi \to \left(\left(\eta \wedge \psi \wedge \theta \right)\leftrightarrow \left(\eta \wedge \chi \wedge \tau \right)\right)$$