I remember trying that line as a kid. My parents proved they didn't HAVE to - I got chores for mouthing off. My mother used to ask us to "cite your source" meaning tell her where you got whatever story/piece of info that you were trying to peddle. Bad source? NOT a good thing.
What does your difficult child not like? Long winded explanations of things? Give him one. Rattle off a 5 paragraph essay on how you, as a parent, do not have to prove anything. You have the right to believe or not believe anything your child says and you are the SOLE authority and judge and jury in your home - so if he continues to mouth off with that phrase he is going to regret it because he won't have X or Y or Z and there is NOTHING he can do about it because you don't HAVE to prove it and if he has a problem with it you can strip his room so he has nothing and he can spend all day every day in there until he can speak to you and the rest of the family with respect.
NOT saying to do that, or say it if you won't, but you do NOT have to put up with that line. You can come up iwth ANY response you want, whether it makes sense or not, because your home is not a democracy or a courtroom. just because the country is a democracy doesn't mean your home is and just because things have to be proven in court doesn't mean you have to prove anythng but that you will follow through with anything you threaten (and you have to follow through because otherwise no kid would behave, kwm?)
If he treasures ice cream, each time he says that you and the other kids get ice cream and he doesn't because he can't PROVE that he didn't do whatever and because he got mouthy.
Wash his mouth out with soap if you want to give it a try.
You can do ANYTHING other than what CPS would consider abuse and YOU don't need proof.
But I would start demanding that HE come up with proof for anything he says/thinks he should have/claims to have done. Simply because he is challenging you to prove things. Make it bite him in the tushie, so to speak. Heck, I might let his siblings do things to mess with him and when he complains or goes off about it, make HIM prove it. And of course, because he claims YOUR word isn't enough then his word isn't CLOSE to enough. Know what I mean??