You need to add more propositions, which may not be accepted. You suggested:
(1) A does not exist
(2) B exists
But (1) has nothing to say about (2) and vice versa, so you need to add another proposition. Perhaps:
(3) A or B exists
If you could show (1) is correct, then (2) is also correct via (3). But you can't prove (1) if (2) is correct unless you assert something like:
(4) Either A or B exists, but not both
This is the exclusive "or", which is much harder to show than the usual inclusive "or" found in (3). Binary choices are common in artificial environments (such as computers), but are more difficult to assert in cases where binary choices are not common. In the real world, it's harder to assert something like: either God exists or evil exists, but not both. It's not immediately clear that propositions in the form (4) are to be preferred over propositions of the form (3). Intuitively, we'd assume the reverse.
You also brought up the statement: "Nonexistence can never be proven." That can trivially be shown to be false. A standard counterexample would be the existence of a married bachelor, which is false by definition. Another example: I don't have at least a million dollars in my bank account and I can prove it. Or: I don't have a best selling book that I've written or a tattoo that says "Mom" on my arm. So you'd need to add some qualifications to that statement to make it true.
If you buy into inductive logic, flying horses do not exist because their is no evidence for them. We can never be 100% sure of that statement because a single counterexample would invalidate all other evidence, but we can be mostly certain which is good enough for most purposes.
In summary, if you convert a claim of nonexistence into a claim of existence, you must take on the burden of proving the premises you used to do the conversion in addition to proving the new claim. In some cases, the extra burden in not worth the effort.