  We consider the contact process on a countable-infinite and connected graph
of bounded degree. For this process started from the upper invariant measure,
we prove certain uniform mixing properties under the assumption that the
infection parameter is sufficiently large. In particular, we show that the
projection of such a process onto a finite subset forms a process which is
$\phi$-mixing. The proof of this is based on large deviation estimates for the
spread of an infection and general correlation inequalities. In the special
case of the contact process on $\mathbb{Z}^d$, $d\geq1$, we furthermore prove
the cutoff phenomenon, valid in the entire supercritical regime.
