How do I get the conditional CDF of $U_{(n-1)}$?

Let $U_1$, $U_2$ .. $U_n$ be identical and independent random variables distributed Uniform(0, 1). How can I find the cumulative distribution function of the conditional distribution of $U_{(n-1)}$ given $U_{(n)} = c$? Here, $U_{(n-1)}$ refers to the second largest of the aforementioned uniform random variables.

I know that I can find the unconditional distribution of $U_{(n-1)}$: It's just Beta(2, $n - 2 + 1$) or Beta(2, $n-1$) because the ith order statistic of uniforms is distributed Beta(i, $n - i + 1$). However, how do I find the conditional CDF of $U_{(n-1)}$ given that $U_{(n)} = c$??

To condition on $A=[U_{(n)}=c]$ is to condition on the event that one value in the random sample $(U_k)_{1\leqslant k\leqslant n}$ is $c$ and the $n-1$ others are in $(0,c)$. Thus, conditionally on $A$, the rest of the sample is i.i.d. uniform on $(0,c)$. In particular $U_{(n-1)}\lt x$ means that the $n-1$ values are in $(0,x)$, which happens with probability $(x/c)^{n-1}$. The conditional density of $U_{(n-1)}$ is $$f_{U_{(n-1)}\mid A}(x)=(n-1)c^{-(n-1)}x^{n-2}\mathbf 1_{0\lt x\lt c}.$$ Thus, conditionally on $U_{(n)}$, $U_{(n-1)}$ is distributed as $\bar U_{(n-1)}\cdot U_{(n)}$, where $\bar U_{(n-1)}$ is independent of $U_{(n)}$ and distributed as $U_{(n-1)}$.