Imagine a scenario where it can be proven that $\forall x \in U\; P(x),$ but only so by contradiction.
That is, assuming $\exists x \in U \; \neg P(x)$ leads to a contradiction, therefore we are forced to conclude $\forall x \in U\; P(x).$
But we just don't have any constructive, or even known, example of an $x \in U$ for which $P(x)$ holds.
"Non-Constructive Case" in the sense that we, till now, don't even now how to construct such an $x.$
Would be great to have an example of mathematical scenario where this happens ! and where, preferably, $U$ is not the empty set, for that would be just trivial vacuous truth.