Let $N_t$ be a renewal process and $B_t=t-S_{N_t}$ with $S_{N_t}=X_1+...+X_{N_t}$. If $F$ is the distribution function of $X_1$, show that $$\int_0^tE(B_t\mid X_1=x)F(dx)=\int_0^tE(B_{t-x})F(dx)$$
My proof:
\begin{align}\int_0^tE(B_t|X_1=x)F(dx)&=\int_0^t E\left(t-(x+\dots+X_{N_t})\right)F(dx)\\[0.2cm]&=\int_0^tE\left(t-x-(X_2+\dots+X_{N_t})\right)F(dx)\\[0.2cm]&=\int_0^tE\left(t-x-S_{N_{t-x}}\right)F(dx)=\int_0^tE\left(B_{t-x}\right)F(dx)\end{align}
Is it correct?
Thank you