  In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for
the $j$-function and, with Mok, have recently announced a proof of its
generalization to any (pure) Shimura variety. We refer to this generalization
as the hyperbolic Ax-Schanuel conjecture. In this article, we show that the
hyperbolic Ax-Schanuel conjecture can be used to reduce the Zilber-Pink
conjecture for Shimura varieties to a problem of point counting. We further
show that this point counting problem can be tackled in a number of cases using
the Pila-Wilkie counting theorem and several arithmetic conjectures. Our
methods are inspired by previous applications of the Pila-Zannier method and,
in particular, the recent proof by Habegger and Pila of the Zilber-Pink
conjecture for curves in abelian varieties.
