Proof of PNT using RH

Question

Is there any proof of Prime Number Theorem that follows immediately from the Riemann Hypothesis? I know that if RH is true then $\sum_{n \leq x} \mu(n) = x^{1/2 + \epsilon}$ for every $\epsilon > 0$. This in turn implies that $\sum_{n \leq x} \mu(n) = o(x)$ from which PNT follows. But I am asking for a proof that follows immediately not via some long route.