Proving a Negative

Fairly regularly, I read on the web people stating emphatically that:

"You can't prove a negative!"

Why not? If A is provable, and we let B=~A, then clearly the negative ~B is provable, because ~B=~(~A)=A.

It is also possible to prove “universal negatives” deductively, depending on the universe. For example, it is possible to prove that there does not exist a solution to exp(x)=0 in the universe of the real (or complex) number system.

As a non-mathematical example, the negative statement “there does not exist a two ton elephant in my dishwasher” seems pretty provable to me too, where “pretty provable” might mean there is a very high likelihood. I could simply weigh the dishwasher along with its contents. If it weighs less than two tons, then I think that pretty much proves it.

Surely some statements are more easily proved than others. "I did not sneeze yesterday" is tough to prove. But so is "I sneezed yesterday." Provability has nothing to do with whether or not the statement is "positive" or "negative." It is not even clear to me what it means for a statement to be "positve" or "negative".