Is there a known theorem $T$ in $ZF+DC$ (or $ZF$ or $ZFC$) such that the only proof we know of $T$ is by using the LEM applied to $A$ ( "$A$ or not $A$" ), where $A$ is *independent* of $ZF+DC$ ? 



Is there a theorem whose only known proof uses "$A$ or not $A$" for undecidable $A$
Here is an example:



> 
> It is provable from $\sf ZF$ that there exists four infinite cardinals, $\frak p,q,r,s$ with $\frak p<q, r<s$ such that $\frak p^r=q^s$. (Here cardinals do not mean just finite ordinals and $\aleph$ numbers.)
> 
> 
> 


[You can find the proof here.](http://karagila.org/2013/provable-equality-of-exponentiation/)


The proof using the axiom of choice is easy, and the proof not using the axiom of choice begins by using the fact that there is a set which cannot be well-ordered in order to construct the example.


Of course, the axiom of choice is indepedendent of $\sf ZF$. I also don't know an explicit proof of this theorem (in fact, before seeing this in a a math.SE question, I don't know if someone had proven this outside of a $\sf ZFC$ context either).


