L\'evy's approach to Brownian local times is used to give a simple derivation of a formula of Borodin which determines the distribution of the local time at level x up to time 1 for a Brownian bridge of length 1 from 0 to b. A number of identities in distribution involving functionals of the bridge are derived from this formula. A stationarity property of the bridge local times is derived by a simple path transformation, and related to Ray's description of the local time process of Brownian motion stopped at an independent exponential time.