For $B$ a Brownian motion and $\Phi\in\Lambda_\text{loc}^2$, do we have for all stopping times $\tau$ $$ \left(\int_0^\cdot \phi_s \, dB_s\right)^\tau = \int_0^\cdot \phi_s \, dB^\tau_s?$$
This question should make sense as $B^\tau$ is an Ito-process and integration w.r.t. to Ito processes is defined.