"Standard theorem on differential equations" that implies a function is analytic

I am reading a proof for a theorem and in the proof we make use of a function $w_z(t)$ defined by the initial value problem

$dw_z(t)/dt = -phi(w_z), w_z(0) = f(z)$

where $z in mathbb{D}$, the complex unit disk and f is an analytic function in the disk. From this the author mentions that a standard theorem on differential equations implies that $w_z(t)$ is analytic in $z$ for fixed $t geq 0$. I am wondering, which theorem is he refering to?