The following assertion appears plausible to me: Let $f(z,w,u)$ and $g(z,w,u)$ be holomorphic in $z$, $w$, and $u$. Suppose that $f(z_0,w_0,u_0) = g(z_0,w_0,u_0) = 0$ and that $f$ and $g$ are non-degenerate in the sense that none of $f(z,w_0,u_0)$, $f(z_0,w,u_0)$,...,$g(z_0,w_0,u)$ are identically zero. Then there exists $w(u)$ and $z(u)$ defined near $u_0$ such that $f(z(u),w(u),u) = g(z(u),w(u),u) = 0$. Note that I am not imposing any uniqueness or regularity requirements on $z(u)$ and $w(u)$.

Is this true? What is a good reference for this?