I am currently trying to write (and prove) a statement and I am having some trouble figuring out if the statement needs to be of the form \begin{gather} p\Rightarrow(q\Leftrightarrow r) \end{gather} or of the form \begin{gather} q\Leftrightarrow (p\wedge r) \end{gather} Using truth tables, one can see that see that the truth values of these statements only differ in the following two scenarios (sorry for the quality of the handwritting):
Further, suppose that I can show (or know) that \begin{gather} \neg p\Rightarrow\neg q\quad\text{ (i.e., }q\Rightarrow p\text{)} \end{gather} Can I then rely on this fact to con conclude that my statement is of the following form? \begin{gather} q\Leftrightarrow (p\wedge r) \end{gather} My intuition is that since $\neg p\Rightarrow \neg q$, the fact that $p\Rightarrow(q\Leftrightarrow r)$ is true when $p$ is false and $q$ is true, the statement $p\Rightarrow(q\Leftrightarrow r)$ cannot be the form of my statement.
More generally, I guess that my question is: how to tell statements $(p\Rightarrow(q\Leftrightarrow r))$ and $(q\Leftrightarrow (r\wedge p))$ apart?
Thank you all very much for your time.