Theorem 3anbi3d

Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006)

	Ref	Expression
Hypothesis	3anbi1d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	3anbi3d	$$⊢\phi \to \left(\left(\theta \wedge \tau \wedge \psi \right)\leftrightarrow \left(\theta \wedge \tau \wedge \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3anbi1d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		biidd	$$⊢\phi \to \left(\theta \leftrightarrow \theta \right)$$
3	2 1	3anbi13d	$$⊢\phi \to \left(\left(\theta \wedge \tau \wedge \psi \right)\leftrightarrow \left(\theta \wedge \tau \wedge \chi \right)\right)$$