How to prove the limit of the empirical distribution function?

Question

I am reading about Exchangeability and de Finetti's theorem and the example in slide 11 confuses me. Consider $(X_i)_{i\in\mathbb{N}}$ with $\forall n \in \mathbb{N},\ X_1,\cdots,X_n \sim \mathcal{N}(0,\Sigma)$ with $$\Sigma_{ii} = 1,\ \Sigma_{ij} = \rho \neq 0,\ for\ i \neq j$$ I wonder how to prove that $$F(x) = \lim_{n\rightarrow\infty}\frac{\sum_{i=1}^n\chi_{(-\infty,x]}(X_i)}{n}=\Phi\left(\frac{x-Y}{\sqrt{1-\rho}}\right)$$ where $$Y=\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^nX_i}{n}$$ I don't know how to start with it since $\chi_{(-\infty,x]}(X_i)$ and $\chi_{(-\infty,x]}(X_j)$ are not independent.