Here is some nice facts and history.
The Dirchelet theorem is extremely complicated to prove. The proof is beyond almost any number theory book you pick up. Even, Hardy and Wright, they prove the Prime Number Theorem, but they state that the proof is so advanced that even their book has to skip it.
This theorem does has some importances in number theory. The Famous Quadradic Reciprocity law, was first incompletely proven by Legendre, but he relied on the fact that infinitely many prime numbers exist in arithmetic progressions satisfing the Dirichlet criterion.
Dirichlet is one of my favorite mathemations, it can be argued, that he was the first to intoduce analysis to number theory. The idea that Dirichlet had is to define a function called "Dirichlet L-function", (similar to the zeta function) and through it he arrived at his result. I believe this proof is from functional analysis (I am sure you heard of it). A strange approach but he managed to do it.
Yes, specific cases, like you mentioned, can be proven rather simply. But the general case is completely different.