optional stopping for randomized stopping time

Question

This is a question about the proof of the Skorohod embedding theorem from Steele's stochastic calculus text. On page 74, given finite sets of nonnegative values $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$, and two integer-valued RVs $I$ and $J$ taking values in the sets of indices, a randomized stopping time is defined as $$\tau=\inf\{t:B_t\not\in(-a_I,b_J)\}.$$ $B_t$ is standard brownian motion. In the course of the proof, the following equivalence is given:$$P(B_\tau=-a_i\mid I=i,J=j)=\frac{b_j}{a_i+b_j}.$$ This looks like the usual hitting time formula for the level of a BM, $P(B_{\tau_{ab}}=a)=\frac{b}{a+b},$ where $\tau_{ab}$ is the first hitting time of $-a$ or $b$. Except that here is is conditional on $I=i,J=j$. Why does this result hold conditionally on $I,J$? The non-conditional result is usually proven using the martingale stopping theorem, is there some extension?