Let $G$ be a connected semisimple Lie group with finite center. Let $(\pi,V)$ be an admissible representation on a Banach space $V$. Is it true that the following are equivalent?

(a) $\pi$ is irreducible

(b) the $({\mathfrak g},K)$-module of $K$-finite vectors is simple.

I can prove this for unitary representations. If the answer is positive, where do I find a simple proof? Say, a proof that does not use the classification of admissible $({\mathfrak g},K)$-modules?