Measurability of Conditional Expectation wrt Stopping Time $\sigma$-Algebra

Question

Suppose that $\sigma$ and $\tau$ are stopping times. Is it true that $\mathbb E[X_\tau\mid\mathcal F_{\sigma}]$ is $\mathcal F_{\sigma\wedge\tau}$-measurable? Here $\mathcal F_\sigma$ and $\mathcal F_{\sigma\wedge\tau}$ are the stopping time $\sigma$-algebras and $X$ is a stochastic process.