In a Bayesian network, does the removal of an edge ever remove existing conditional independences?

Question

I am wondering if the removal of any edge in a acyclic Bayesian network ever removes an existing conditional independence?

Intuitively, I would think not, but I was wondering if there is a formal proof for this?

For example:

A -> B -> C

Then we have $A \perp C | B$, but if we remove any of the edges, we would still have $A \perp C | B$.