Positive scalar curvature and the Euler class

We prove the following generalization of the classical Lichnerowicz vanishing theorem: if $F$ is an oriented flat vector bundle over a closed spin manifold $M$ such that $TM$ carries a metric of positive scalar curvature, then $<\widehat{A}(TM)e(F),[M]>=0$, where $e(F)$ is the Euler class of $F$.