There is an interesting class of problems that are provable in $ZF$ by "coincidence" in the sense that either (a fragment of) $AC$ holds and then it is provable from the proof in $ZFC$, or there specific instance of the failure of $AC$ implies the result.
The simplest example for such problem is: countable product of spaces with $2$ elements is compact.
If the product of our spaces is non-empty, we have fragment big enough of $AC$ to prove that the product is compact by following the proof in $ZFC$, otherwise the product is empty, but the empty space is compact, so we are done.
A more complicated example is: let $X$ be subspace of $ℝ$, then $X$ is compact if and only if $X$ is sequentially compact and Lindelöf.
If $CC(ℝ)$ holds then the usual proof works, on the other hand $¬CC(ℝ)$ is equivalent to "subspace of $ℝ$ is Lindelöf if and only if it is compact", so because $X⊆ℝ$ is Lindelöf, it is compact.
The proof that $¬CC(ℝ)$ is equivalent to "Lindelöf ⇔ Compact" for subspaces of $ℝ$ is, as far as I know, not a trivial result.