Composition with Cumulative Distribution Functions

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Let's say, I had two independent random variables $\epsilon_1,\epsilon_2$, then I could express the joint CDF by

$$F(\epsilon_1,\epsilon_2)=F(\epsilon_1)F(\epsilon_2)$$

where F is a cdf of my choice and use the independence Coppula.

Now, let's say, I had two random variables $x_1,x_2$ with

$$\begin{align}x_1&=b+\epsilon_1,\\ x_2&=Ax_1+b+\epsilon_2.\end{align}$$

How can I express the cumulative distribution $F(x_1,x_2)$ using terms from $F(\epsilon_1,\epsilon_2)$?