The more general question is, when is anything a proof.? There are a least two answers, but the one I think is relevant here is that something is a proof if it is a persuasive argument that someone skilled in the art will find convincing. In this sense, your finite enumeration is a proof, if:
- There is a persuasive argument that your enumeration of the cases is in fact exhaustive
- The argument for each case is persuasive
There was a lot of controversy in the 1980s about the Haken-Appel proof of the four-color map theorem, which states that that every map in the plane can be colored with only four colors so that no adjoining regions are the same color. Haken and Appel had an argument that showed that every map could be reduced to one of a few thousand cases, an argument that showed that if a case satisfied certain conditions then the corresponding maps could be four-colored, a computer enumeration of the several thousand cases, and a computer-generated demonstration that each case had the desired property.
The arguments were checked and now everyone agrees that this was a proof. But for a while it wasn't clear.