"hold up" can be used in several situations. "I'm driving across the country, I hope my car can hold up!" - means that he hopes his car will survive the trip and not break down. "She's not asking a lot of questions, I think my lie is going to hold up." - means that he thinks she will not discover his lie because she isn't asking enough questions to discover the truth. "That won't hold up in court" - means that the legal argument you are using will not survive the counter-argument in court. One lawyer may say "I'm going to call a witness to the crime who will say your client robbed the bank" and the other lawyer may respond "that won't hold up in court because my client was on an airplane that afternoon and I can prove it!". "That will hold up in court" - means that the legal argument you are using will survive the counter-argument in court. One lawyer may say "I'm going to claim that my client is innocent of the crime of theft, and that will hold up in court, because he took the radio out of your clients garbage can, and the law states that once an item is in the garbage can at the side of the road it is no longer the personal property of the person who put it there."

Your lawyer will make your case. He will make an argument. The other lawyer will try to knock down that argument. You hope that argument will hold up.

US: In trials, there are specific rules about what types of evidence are allowed. There are some statements or even physical objects that cannot be mentioned during the trial because it would be unfair. There is also often some information that is so uncertain that the opposite side will have an easy time proving it less than true. So if something "will hold up in court" it means that a lawyer could use that evidence in a trial successfully. Metaphorically, people will use the phrase to say that something could be proved true. (or the opposite if you can't prove it). For example, "I think that Bill probably messed up the accounting - it wouldn't hold up in court, I just have a feeling that something is wrong."

held up in court = will be accepted by the legal system (e.g. that video won't hold up in court because you obtained it illegally!!)

held up in court = will be accepted by the legal system (e.g. that video won't hold up in court because you obtained it illegally!!)