Yes.
It is an excercise for you (unless you want me to prove it) that the set G_n={1<=x<=n |gcd(x,n)=1} for n>1 is a group under multiplication modulo n. (That is the set of all relative prime elements to n less than n is a group). Since it is finite we have a^m = 1 where m is the number of elements in the group. That is m=phi(n) by definition.
Q.E.D.
Now, this group is isomorphic to Z/nZ (you know the classes of integers modulo n) thus, this is true for all integers relatively prime to n.