Even if you are able to somehow prove that the landlord is fully responsible for the utility bills, it won’t exactly release you from your contractual obligations of paying rent. The landlord will be able to prove to the courts that you are in violation of the lease terms, thus holding you fully responsible.

The law doesn’t allow you to withhold rent for any reason. While I may agree with you that you should have had the AC unit fixed immediately; the courts will not see it that way. All they can do is look at the facts. The facts are that you had an agreement to pay rent on a specific date and did not.

You could try to use the RPA complaint option to provide you with some additional legal documentation, that may help the courts see that the problem was more than just an excuse to pay rent. That’s up to you.