  Given a non-positively curved cube complex $X$, we prove that the quotient of
$\pi_1X$ defined by a cubical presentation $\langle X\mid Y_1,\dots,
Y_s\rangle$ satisfying sufficient non-metric cubical small-cancellation
conditions is hyperbolic provided that $\pi_1X$ is hyperbolic. This generalises
the fact that finitely presented classical $C(7)$ small-cancellation groups are
hyperbolic.
