  Recent work showed that a theorem of Joris (that a function $f$ is smooth if
two coprime powers of $f$ are smooth) is valid in a wide variety of
ultradifferentiable classes $\mathcal C$. The core of the proof was essentially
$1$-dimensional. In certain cases a multidimensional version resulted from
subtle reduction arguments, but general validity, notably in the quasianalytic
setting, remained open. In this paper we give a uniform proof which works in
all cases and dimensions. It yields the result even on infinite dimensional
Banach spaces and convenient vector spaces. We also consider more general
nonlinear conditions, namely general analytic germs $\Phi$ instead of the
powers, and characterize when $\Phi \circ f \in \mathcal C$ implies $f \in
\mathcal C$.
