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 result of $¬CC(ℝ)$ being equivalent to "Lindelöf ⇔ Compact" for subspaces of $ℝ$ don't, as far as I know, have a simple proof.

> <cite authors="Herrlich, Horst">_Herrlich, Horst_, [**Products of Lindelöf \(T_2\)-spaces are Lindelöf – in some models of ZF.**](http://www.emis.de/journals/CMUC/cmuc0202/cmuc0202.htm), Commentat. Math. Univ. Carol. 43, No. 2, 319-333 (2002). [ZBL1072.03029](https://zbmath.org/?q=an:1072.03029).</cite>