Scope of lexical binding

Aris asked a very good question: why do we write proofs as "Let x in X...[good thing happens] Since x was arbitrary, we've proven that for all x in X,..." rather than simply "For all x in X, [good thing happens]" ?

I think it has to do with lexical scoping.
When we say "for all x in X", that typically ties up the letter x to mean a certain something until the end of the sentence, or possibly the paragraph. But no further.
For example, one may say "for all i in 1,2,3,...,n" in every sentence, and have i free to mean something different each time.
But when we start out a proof with "Let x be in X" then the reader may assume that x will mean the same fixed element, for the rest of the proof.

Mind you, we're not discussing mathematics here; we're only discussing how mathematics is written and read.
And it's not to say that it isn't prone to error. All the time, in lectures, one hears the speaker (or the audience) say "where this n is not that n" or even just "where n doesn't equal n, of course"