I originally posted this on math.stackexchange, but then deleted it and moved it here since I think it would fit this site more.
I saw a claim in a slideshow from a basic computer architecture course that given some boolean operator $T(x_1,...,x_n)$, if it is universal then necessarily: $$T(x,...,x)=\neg x$$ However, I found no explicit mention of this claim anywhere else, including here. Is it correct? And if so can one please provide a proof or a proof-idea? Because it was provided as a fact completely unrelated to the rest of the slides I have no idea how to approach it, and I am not sure I even have the right tools to do so.
I have already seen this question, for example, but there the proof is tailored towards a specific function. My problem is with proving the general case.