We do not know this for absolute certain, but when we study the light from far away galaxies we see similar kinds of spectral lines as we see from elements close to us on the Earth. The relationships among the spectral lines in distant atoms also follow the same rules they do here, so the laws of atomic physics, and the value of fundamental constants that determine these physical laws seems to be the same everywhere. As for the speed of light being different in some galaxies, this would lead to a very complicated universe and there is no evidence that the universe is anything but the rather uniform thing that it is with laws and constants being the same everywhere we can look.