Theorem anbi1d

Description: Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 16-Nov-2013)

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

Proof

Step	Hyp	Ref	Expression
1		anbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	a1d	$$⊢\phi \to \left(\theta \to \left(\psi \leftrightarrow \chi \right)\right)$$
3	2	pm5.32rd	$$⊢\phi \to \left(\left(\psi \wedge \theta \right)\leftrightarrow \left(\chi \wedge \theta \right)\right)$$