One of the laws of logic, the law of non-contradiction, is in fact a provable negative. A proposition cannot simultaneously be true and false. If you've ever taken a course in logic you've probably formally derived this law using the concept of the empty (null) set and the valid rules of inference (if not, you can read the proof for yourself in book on formal logic). Furthermore, upon doing this you will have formally demonstrated that it is not the case that one of the laws of logic is that you can't prove a negative, thereby proving a negative in the process of deriving the law itself.

Also, just to drive this home, the phrase "you can't prove a negative" is a negative statement. Therefore, if you could show it to be a true statement then it would by definition actually be false. The only alternative is that the law of non-contradiction is invalid, but if that's the case then logic as we know it can't exist. Which one of these two possibilities seems more likely to you?