Let $(X_t)_{t \geq 0}$ and $(Y_t)_{t\geq 0}$ be stochastic processes where $X_t$ and $Y_t$ are continuous random variables for all $t \geq 0$ (with continuous densities). Let $T$ be also a continuous random variable. Assume that
$P(X_t \in B | Y_t) = \int_B f(x,Y_t,t)\, dx $
for all measurable sets $B$ and with a continuous density $f$.
Is it true that $P(X_T\in B | Y_T, T=t) = P(X_t \in B| Y_t)$ for $t \geq 0$? Intuitively, it should be true and I would like to prove it using a similar approach like here, i.e. with conditioning on $\{ t \leq T \leq t+\epsilon \}$ instead of $T=t$ and then letting $\epsilon \rightarrow 0$, but I'm having trouble with the dependencies of $X_T$ and $Y_T$ on $T$ and also with how to handle conditioning on $Y_T$ and $T$ at the same time and if a joint density of $X_T, Y_T, T$ or something similar is also needed.
Any help is appreciated! :)