Will most rain boots also work well in snow and ice?

I’m moving out of state for school next month and am looking for some rain boots. Where I’m going is also notorious for ice and snow in the winters, so I was wondering whether or not I should get some boots especially for snow and ice, or if I could buy some rain boots and let them multifunction.
(I’ll be living on campus, which means I’ll be walking quite a lot for classes)

Thanks!