Characteristicity is strongly intersection-closed for any variety of algebras

Statement
Suppose $$\mathcal{V}$$ is a variety of algebras and $$A$$ is an algebra in $$\mathcal{V}$$. Suppose $$B_i, i \in I$$ are all characteristic subalgebras of $$A$$. Then, the intersection $$\bigcap_{i \in I} B_i$$ is also a characteristic subalgebra of $$A$$.