While she may not be able to take the AC system legally there are several issues to consider e.g. it may be difficult stopping her from taking major pieces of the system, or destroying it by smashing and bending everything.
A good AC system is critical in Florida and I would like to hear more-complete details from both sides. Did she notify the brother-in-law that her AC was broke and did he offer or refuse to pay for the system since he is the property owner and most-likely it is his responsibility to repair or replace the system. You can't put blame on the tenant in regards to whether the tenant repaired or replaced the system since the owner may have left that choice up to her. Since some AC refrigerants are now illegal to use it is absolutely necessary to replace entire systems.
It the tenant paid $6,000 to install the furnace only a few months ago then the tenant may have a chance of suing the owner to recover her money and if a court favors the tenant then the owner is wrong and should have done what was right before it went to court.
I've owned rental properties for more than 50 years and there has been a few times that tenants did repairs without my knowledge and then sent me the bill. Of course, I sometimes get excited for not being told and because the tenant paid two times what I could get the work done for. I also know that if the tenant takes me to court I will most-likely lose and will have to pay the high price the tenant paid.
One of the biggest issues I questions is the owner's morals even if the tenant did not give the owner an advance notice. The owner should not snicker and walk away with a free AC system if the system was broke and regardless of whether or not it could have been repaired. Now, the owner has a new system that should last no less than 40 years and why should he earn a larger profit because the tenant used her good faith and hard-earned money to pay for the system even when she was not the property owner.
$6,000 is a very reasonable price for a complete AC system changeout. The least the owner should offer to pay the tenant for the system is $4500. I don't know the whole story, but it looks like the tenant should take the owner to court and I will bet she wins.
For this situation the owner should be honest within himself, do what is right, sit down with the tenant and discuss everything without badgering to tenant so he can come out of this smelling like a rose and walking away with something he should have paid for in the first place.