This result holds less obviously for Brownian motion with constant drift, not just $0$ drift. It is critical that the starting point is centered on the interval and it fails otherwise.

Stern, F. An Independence in Brownian Motion with Constant Drift. The Annals of Probability, Vol. 5 (1977), 571-572.

This holds for biased random walks because reflecting the paths to one boundary point gives paths to the other boundary with a constant magnification of probability. Taking the limit shows that the same is true for Brownian motion with constant drift.

This result holds less obviously for Brownian motion with constant drift, not just $0$ drift. It is critical that the starting point is centered on the interval and it fails otherwise.

Stern, F. An Independence in Brownian Motion with Constant Drift. The Annals of Probability, Vol. 5 (1977), 571-572.

This holds for biased random walks because reflecting the paths to one boundary point gives paths to the other boundary with a constant magnification of probability. Taking the limit shows that the same is true for Brownian motion with constant drift.