It is a standard result that if $W_t$ is a Brownian Motion and $S$ is a stopping time of the standard filtration $F_t$ then we have that $B_t = W_{S+t} - W_S$ is a Brownian Motion.
I quote the theorem above because I think it might be useful for showing the following:
I am trying to show that $W_{t-T_b}$ is symmetric in the sense that $P^{b}[W_{t-T_b} \leq a] = P^b[W_{t-T_b} \geq 2b -a]$ on $\{T_b \leq t\}$, where $T_b$ is the first hitting time of $W$ to $b$.
Any help would be appreciated. Thanks.