homework and exercises - How to show that every Killing vector field is a matter collineation?

Various texts make this claim, but no proof is given.

Explicitly, let $L$ denote the Lie derivative. Suppose $L_X g_{ab} = 0$ for some vector field $X$, called a Killing vector field. Suppose that the stress-energy tensor $T_{ab}$ satisfies the Einstein field equations. Then, show that $L_X T_{ab} = 0$.