No, that's not the way it works.
First, anybody here is a Negative Polarity Item, triggered by the negative in don't think. So it's just part of the machinery, a constant
X set to zero. Putting it there as the subject of the complement clause links the clause to the negative in the higher clause, which leads to the next point.
Let's dispose of the any and just substitute a noun to illustrate the structures, OK?
Second, think is what's called a Negative-Raising verb, like believe (but unlike know or claim). Neg-Raising is the phenomenon that relates (for instance)
- Bill thinks that Mike is not coming.
- Bill does not think that Mike is coming.
The negation of think in the first is intended and interpreted as a negation of be coming, as in the second. So the two sentences are synonymous. Neg-Raising is governed by think, and believe, but not by know or claim, since the second examples are not synonymous:
- Bill doesn't think/believe Mike is coming. = Bill thinks/believes Mike isn't coming.
- Bill doesn't know/claim Mike is coming. ≠ Bill knows/claims Mike isn't coming.
This is the more normal situation; Neg-Raising is a minor rule, governed by only a few mental predicates. We say that these predicates are "transparent to negation", since a reading of outside negation with think or believe (like there is with know and claim) is very hard to produce, and generally requires very careful stress, intonation, and rhythm to accomplish successfully.