Prove decomposition formula for $E^x(f(X_t);\tau_0<t)$, where $\tau_0$ denotes the first hitting time of $0$

Question

Let the strong Markov process $(X_t)_{0\le t\le T}$ be the unique week solution to the SDE

$$dX_s=\sigma(X_s)dB_s+CdL_s^0(X)$$ $$X_0=x \quad P^x-a.s.\quad(x>0)$$

Let $\tau_0$ be the first passage time of the process $(X_t)$ at point 0 $$\tau_0(X)=inf\{s>0,X_s=0\}\land T$$

Let $r_0^x$ be the density of $\tau_0$ under the probability $P^x$, I need to proove that $$ E^x(f(X_t)\mathbb{1}_{\tau_0<t})=\int_0^tE^0f(X_{t-s})r_0^x(s)ds $$