  We generalise Dwork's theory of $p$-adic formal congruences from the
univariate to a multi-variate setting. We apply our results to prove
integrality assertions on the Taylor coefficients of (multi-variable) mirror
maps. More precisely, with $\mathbf z=(z_1,z_2,...,z_d)$, we show that the
Taylor coefficients of the multi-variable series $q(\mathbf
z)=z_i\exp(G(\mathbf z)/F(\mathbf z))$ are integers, where $F(\mathbf z)$ and
$G(\mathbf z)+\log(z_i) F(\mathbf z)$, $i=1,2,...,d$, are specific solutions of
certain GKZ systems. This result implies the integrality of the Taylor
coefficients of numerous families of multi-variable mirror maps of Calabi-Yau
complete intersections in weighted projective spaces, as well as of many
one-variable mirror maps in the "Tables of Calabi-Yau equations"
[arXiv:math/0507430] of Almkvist, van Enckevort, van Straten and Zudilin. In
particular, our results prove a conjecture of Batyrev and van Straten in [Comm.
Math. Phys. 168 (1995), 493-533] on the integrality of the Taylor coefficients
of canonical coordinates for a large family of such coordinates in several
variables.
