The Proof is of the Form P $\displaystyle \Leftrightarrow$ ... $\displaystyle \Leftrightarrow$ Q thus the proof shows that P $\displaystyle \Leftrightarrow$ Q which means that both P $\displaystyle \Rightarrow$ Q and Q $\displaystyle \Rightarrow$ P are true and thus I don't see why you would have to work it the other way.