Contraposition and law of excluded middle Does truth-equivalence of an $A \rightarrow B$ and contrapositive $\neg B \rightarrow \neg A$ rely on the law of excluded middle?

It would seem that it does not, but rather, on the law of non-contradiction. Assume

1. $A\implies B$
2. $\
eg B$



Now, to obtain a contradiction, suppose $A$. Then we have $B$ from (1). But we have $\
eg B$ from (2), so by the law of non-contradiction, it follows that we have $\
eg A$, and finally we have $\
eg B \implies \
eg A$.

The distinction between this and the law of the excluded middle is subtle but it's there. In order to prove via the law of excluded middle, we would knock out assumption (2), and instead assume $\
eg (\
eg B\implies \
eg A)$, and from here derive a contradiction, yielding $\
eg \
eg (\
eg B \implies\
eg A)$, and by the law of the excluded middle, we would conclude that $\
eg B \implies \
eg A$.

For more on the distinction, see here.