I am a little stymied by the following:$\def\imp{\Rightarrow}$ $(P \imp Q) \land (Q \imp P) \equiv (P \lor Q) \imp (P \land Q)$
Working with the RHS I have:
$\neg(P \lor Q) \lor (P \land Q) $
$( \neg (P \lor Q) \lor P) \land (\neg (P \lor Q) \lor Q))$ by distributive
$((\neg P \land \neg Q) \lor P) \land ((\neg P \land \neg Q) \lor Q)$ by de Morgan's Law
$((P \lor \neg P)\land(P \lor \neg Q)) \land ((Q \lor \neg Q)\land(Q \lor \neg P))$ by distributive
$(P \lor \neg P) \land (Q \imp P) \land (Q \lor \neg Q) \land (P \imp Q)$
Assuming this is correct so far, I am very close to $(P \imp Q) \land (Q \imp P)$, however I am unsure of how to deal with the $X \lor \neg X$ 's in the above, have I made an eror in the above expansions?
\land
and\lor
specifically for logical connectives. $\endgroup$