Cellular homotopy theorem

Statement

Suppose $(X,A)$ and $(Y,B)$ are relative CW-complexes (i.e. CW-pairs). Let $f:(X,A)\to (Y,B)$ be a continuous map. Then there exists a cellular map $g:(X,A)\to (Y,B)$ such that $f$ and $g$ are homotopic relative to $A$ (that is, the homotopy does not change the restriction of the function to $A$ at all).