A "the following are equivalent proof" lists some number of statements and proves that there are if and only if relationships between all of them. For more than two statements there is not a unique way to do this. I am wondering how many implications in total need to be proven for n statements.
If you think of each statement as a vertex and each implication as a directed edge then the problem can be restated as follows:
what is the smallest path connected directed graph on n vertices? (Smallest meaning smallest number of edges)
If you have $n$ statements, you can do it with $n$ implications.
Each statement needs to be the target of at least one implication, so at least $n$ implications are needed.
And you can complete the proof with $n$ implications by proving that statement $i$ implies statement $i + 1 \pmod{n}$ for each $i \in \{1, \cdots, n\}$.
Moreover, this structure—a cycle of implications among the $n$ statements, in some order—is the only minimal strongly connected graph on $n$ vertices. If every vertex has indegree one and outdegree one, then the graph consists of cycles, and if it is connected then it is a directed $n$-cycle. On the other hand, if any vertex has a higher indegree or outdegree, then there must be more than $n$ edges total.
Of course, it may be easier to complete the proof using more than $n$ implications and a less rigid structure.