How did they hid the camera in this mirror shot in “The Best Years of Our Lives” (1946)

I watched this on the big screen for the first time and this take blew my mind. I can’t figure out how they hid the camera in this scene, it starts when they go in the ladies room. There are 5 camera movements switching between 3 mirrors all in one take and I can’t figure out how they hid the camera. My wife thought it was doubles but the actor and the reflections blink at the same time. This might be one of my new favorite shots. It is really good and subtle.