I would also accept this as an axiom. It’s obviously true, but I see no way to prove it. It is not what Spock was referencing in the above quotation, but the content of the quote still applies.
Amazingly, though, I have encountered people who think this is up for debate.
It isn’t.
RobertLovesPi.net 
Logic is the only setting I know of where this has a strict enough definition to be examined with scrutiny. However, in that setting it is provable since x represents a statement that is either true or false and (not true)=false and (not false)=true. Now if we assumed the opposite of your statement true we would always end up with a contradiction and thus your initial statement is true.
In other settings I don’t know of a strict enough definiton of “not x” to examine the statement. It could mean
which is kind of circular and trivially true. “not x” could also refer to the whole set of elements that have the property that they are not equal to x in which case your statement is also true, but still contains virtually no information because it is comparing single variables to sets of variables. Another interpretation is to just consider “not x” as a variable of its own leaving no thought to what “not” usually means. In this case the statement is equivalent to a statement of the type 
and whether it is true or not depends completely on context.
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Seems like symbolic logic is the way to go. (But people can and will debate anything.) 🙂
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