I’m not sure how you prove by negation in this case just via modus ponens. Care to enlighten me? I opened with something that doesn’t follow so that would be a great place to start.
Give me a consistent formal system with a list of theorems to prove OP’s conjecture and I’ll show you how we have gaps in the system. My analytic philosophy is pretty rusty; I think there are a few 20th century folks you can start from for this.

Can you help me understand which political petitions meant to document real constituent desires don’t require doxxing yourself? I don’t believe I’ve ever participated in any citizens initiative that didn’t require personal information.