Consider a statement of this form:
We don't know whether or not there exist any [X] with property [Y], but we do know that [some concrete instance/example of an X] has property [Y] iff any [X] has property [Y].
Is there a name for this sort of statement? When is it possible to (truthfully) make such a statement? Where have such statements appeared in math?
(The one statement I've heard that inspired this question is that we have a (completely impractical) algorithm that will solve any problem in NP, and we know that it will run in polynomial time iff P=NP.)
I also realize this is similar to Unique candidate that fails, except that we'd know of a unique candidate but not know whether or not it fails.