Weaker version of Central Limit Theorem

Question

The Central Limit Theorem applies to i.i.d random variables. I’ve seen mention of a version of the CLT that applies to the sum of random variables that while independent, are not identically distributed that claimed that their mean still tends toward a normal distribution, but I can’t find confirmation that this is true, and if so a formal statement and accompanying proof. Is it generally true that the mean of the sequence of partial sums of independent but not identically distributed random variables converges in probability to a standard normal distribution (or something along these lines; I may have some details wrong), and if so can you point me to a reference, ideally with a proof? Or if it’s not true can you explain why not?