For a Markov process $X_t$, define a stopping time $\tau$ and $P_x(\cdot)=P(\cdot|X_0=x)$, $$P_x(X_t=x, \tau<t)=P_x(X_t=x|\tau<t)P_x(\tau<t)=P_{X_{\tau}}(X_{t-\tau}=x)\mathbb{E}_x\mathbb{1}_{\tau<t}$$
My question is that how to the last equation.
Is the first term that using the strong Markov property and the second term is $$P_x(\tau<t)=P(\tau<t|X_0=x)=\mathbb{E}(\mathbb{1}_{\tau<t}|X_0=x)?$$