Does the Leibniz rule for $\frac{\partial}{\partial x}\int_{a}^{b}f\left(x,y\right)dy$ apply when $x$ is a function of $y$?

Question

Leibniz rule:

$$\frac{\partial}{\partial x}\int_{a}^{b}f\left(x,y\right)dy=\int_{a}^{b}\frac{\partial f\left(x,y\right)}{\partial x}dy$$

But what if $x$ is a function $y$? (Assume it's smooth and has any nice properties you want, but it isn't a constant.) Can I apply Leibniz rule and say

$$\frac{\partial}{\partial x}\int_{a}^{b}f\left(x\left(y\right),y\right)dy=\int_{a}^{b}\frac{\partial f\left(x\left(y\right),y\right)}{\partial x}dy$$