Let $X(t)$, $0\leq t\leq 1$, be a bridge from $0$ to $0$ with exchangeable increments on $D[0,1]$. We obtain the n.a.s.c. for the sojourn below $0$ to be uniformly distributed, or equivalently for $X$ to have a uniform index of the (unique) supremum. This is applied to Lévy bridges.