Contour Integration by Substitution

Theorem
Let $f$ be a holomorphic function of a simply-connected region $V \subseteq \mathbb C$.

Let $\gamma_V$ be a contour in $V$ connecting $v_1$ to $v_2$.

Let $U$ be a region.

Let $\gamma_U$ be a contour in $U$ connecting $u_1$ to $u_0$.

Let $u: U \to V$ be a holomorphic function with $u^{-1}\left[ {\{v_0,v_1 } \right\]\ne \varnothing$.