Let $F(u,v)$ be a continuous bivariate cdf, $F(u|v)$ the conditional distribution of U given V=v and $F^{-1}(s|v)$ the corresponding inverse conditional distribution. Let $G$ be a continuous cdf and $(\epsilon_1,\epsilon_2,\ldots)$ be a sequence of iid U(0,1) random variables. A 1-dependent series $Y_t$ with stationary distribution G is defined as follows: $Y_t = G^{-1}[F^{-1}(\epsilon_t|\epsilon_{t-1})]$.
The joint distribution of $(Y_{t-1},Y_t)$ is given as follows:
$ P(Y_{t-1}\leq x,Y_t\leq y) = Pr(\epsilon_{t-1}\leq F(G(x)|\epsilon_{t-2}), \epsilon_t\leq F(G(y)|\epsilon_{t-1}))$ $= \int_0^1\int_0^1P(u_2\leq F(G(x)|u_1),\epsilon_t\leq F(G(y)|u_2))du_2du_1 \stackrel{?}{=} \int_0^1\int_0^{F(G(x)|u_1)}F(G(y)|u_2)du_2du_1$ $ = \int_0^1F(F(G(x)|u),G(y))du.$
Source: Joe, H. "Multivariate Models and Dependenceny Concepts".
My question: I do not understand the equality which I marked with an "?" above the equal sign.
Can someone please elaborate on this? What assumptions are envolved?
Thanks!