A statement following from the law of excluded middle Does the statement ~~$A\equiv A$ follow from the law of excluded middle? According to my book which is not on logic it does, but I do not know how to use the law of excluded middle for this simple tautology.

Assume $A$ and $\
eg A$. Then $A\land \
eg A$, which is absurd, hence $\
eg\
eg A$. So $A\implies \
eg\
eg A$: this holds without the law of excluded middle.

Now assume $\
eg\
eg A$. If $A$, then $A$. If $\
eg A$, then $\
eg A\land \
eg\
eg A$ which is absurd, hence $A$.

So whether $A$ holds or $\
eg A$ holds, we always get that $A$ holds : thus [it's this "thus" that uses the law of excluded middle] $\
eg\
eg A\implies A$