Say $X$ is a smooth threefold, and $A$, $B$, and $C$ are three smooth divisors on $X$. Is it true that the three-way intersection $(A \cdot B \cdot C)_X$ is equal to the intersection $(A\vert_C \cdot B\vert_C)_C$ computed on $C$?
Geometrically this seems like it should be true, but I'm having a hard time finding a reference. No doubt it implicit somewhere in Fulton. In what generality does this work? (e.g. I guess it is still true if $A$ and $B$ are not effective, but maybe not if $C$ is too singular)