When proving that \begin{equation} A\iff B \end{equation} We generally split the prof into two parts:
\begin{equation} A\implies B \tag{1} \end{equation} \begin{equation} \tag{2} B\implies A \end{equation} In the cases I have seen, these proofs are completely independent. Today I've come across a different proof, though. After proving $(1)$, it proved $(2)$ by using $(1)$ in the last part of the proof. Is it fine to do so from a logical standpoint?
This is OK. After proving $A\implies B$ you can use it for whatever proof you like,
even for the proof of $B\implies A$. This does not introduce a logical flaw of some sort.