Q:

Proof of multiple equivalences in elementary set theory. 

Can someone give a proof or a formal explanation of how are we justified to use this simpler way of proving equivalences described in below:
As an example, let's say that we want to prove the equivalence of three different statements $\alpha$, $\beta$, and $\gamma$. One sure way to do this is to show that $\alpha \iff \beta$, $\beta \iff \gamma$, and $\alpha \iff \gamma$ which is equivalent to proving six different conditionals.  
But in my textbook I've seen that the author had used a different approach for a similar situation involving the equivalence of three different statements.  
Instead of the way I've shown in above he proved that $\alpha \Rightarrow \beta$, $\beta \Rightarrow \gamma$, and $\gamma\Rightarrow \alpha$. 
One short note: I searched a lot but I couldn't find any similar question to get my answer.  So if you already know a similar question which makes my question a duplicate please inform me to remove my question entirely. 
Thanks. 

A:

Take $\alpha\iff\beta$, for example.  This statement is the same as $(\alpha\implies \beta)\wedge( \beta\implies\alpha)$. The first statement we get for free.  Then, by modus ponens, since $\beta\implies\gamma$ and $\gamma\implies \alpha$, we have $\beta\implies\alpha$.

