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If you have a copy of the notice you provided them, they can say whatever they want, but the proof is in what you have a copy of. You need to send them a letter informing them that you provided the notice in accordance to the 60 day term on the lease and you are attaching a copy of that notice as proof you sent it to them. Tell them it is not your fault if through their negligence they missed the notice, but you are not liable to them for their own negligence. Tell them if they continue to pursue this, you will be forced to take legal action against them for negligence and any damage they try to cause you by pursuing the rent or filing any negative credit reports against you and that you will pursue them for punitive damages as well for taking their actions in bad faith since you have proof you provided them notice and you will also seek attorney's fees against them for that bad faith.
That is all you can do for starters. If they refuse to agree, then your final option in this matter is you have to sue them for breach of the lease and their negligence in failing to properly handle the timely notice you submitted to them in July.