The question might be silly, but I really don't understand, why in most of the books when one states the theorem about the Asymptotic Normality of, let's say ML estimator, there has always been said first about the consistency $$ \hat{\theta_{n}}\stackrel{p}{\to} \theta_{0} $$
and then next about the Asymptotic Normality
$$ \sqrt{n}(\hat{\theta_{n}} - \theta_{0} ) \stackrel{d}{\to} \mathcal{N}\Big(0, \sigma \Big). $$
It is obvious, that in this case the statement about consistency follows from the one about Normality...
Well, if it comes to ML estimator, first we prove the consistency, then next, normality... may be this is why...
I can guess that there are several reasons. Maybe one is the fact that convergence in probability is "stronger" then convergence in distribution and actually the former implies the second. Another reason I can think of is that in the context of estimation, consistency is a point property while limit distribution is mainly serves to build asymptotic CI's. Thus we usually begin with the point properties. Moreover, pretty common case is when the variance function of the normal limiting distribution is decreasing w.r.t. $n$, as such we can say that it converges to a constant and hence implies consistency. So it useful to know what is consistency before the discussion of convergence in law.