Suppose I have $$x^{(c(p-1))} \equiv y^{(p-1)} \pmod{p}.$$ I would like to take the (p-1) root of both sides to get: $$x^c \equiv y \pmod{p}$$
I really just want to know if this a valid technique and what it would take to make it rigorous (I already know to show that $x,y$ are not congruent to 0)?