Statement
If $T$ is a stopping time for Brownian motion, then on $\{T<\infty\}$ the shifted process
$$ (B_{T+s}-B_T)_{s\ge 0} $$is a Brownian motion independent of the information before time $T$.
Interpretation
After a stopping time, Brownian motion starts afresh from its current position.
Use
This is the key tool behind reflection arguments and many hitting-time proofs.