Hi I am trying to prove the following:
Let $\{v^j\}_{j = 1}^n$ be iid random variables $v_j : \Omega \rightarrow [\underline{v}, \bar{v}]$ and let $H(\cdot)$ and $h(\cdot)$ be their cdf and pdf, respectively. Fix an $i \in \{1, \dots, n\}$. Then we have that
$$\mathbb{E} \left[\max_{l \neq i} \{v^l \} |1_{v^l < v^i \ \forall l \neq i} \right] = \int_{\underline{v}}^{v^i} t (n-1) \left[\frac{H(t)}{H(v^i)} \right]^{n-2} h(t) \ dt$$
What I have so far is the following: \begin{align} \mathbb{E} \left[\max_{l \neq i} \{v^l \} |1_{v^l < v^i \ \forall l \neq i} \right] &= \frac{ \mathbb{E} \left[\max_{l \neq i} \{v^l \}, 1_{v^l < v^i \ \forall l \neq i} \right]}{\mathbb{E}(\mathbb{1}_{v^l < v^i \ \forall l \neq i})} \\ &= \frac{ \mathbb{E} \left[\max_{l \neq i} \{v^l \}, 1_{v^l < v^i \ \forall l \neq i} \right]}{\mathbb{P}(v^l < v^i \ \forall l \neq i)} \\ &= \frac{ \mathbb{E} \left[\max_{l \neq i} \{v^l \}, 1_{v^l < v^i \ \forall l \neq i} \right]}{[H(v^i)]^{n-1}} \\ \end{align}