Can distributive law be applied into if-then and only-if statements?

Question

Are the two statements can be logically equivalent, just like in OR, AND expressions?

[(p ^ q) ↔ r] = [(p ↔ r) ^ (q ↔ r)]

[(p ^ q) ⇒ r] = [(p ⇒ r) ^ (q ⇒ r)]

Answer 1

It's a bit of a mess. Some hold and some don't. Below shows you what holds as an equivalence ($\Leftrightarrow$) and what does not ($\not \Leftrightarrow$):

$(P \land Q) \to R \not \Leftrightarrow (P \to R) \land (Q \to R)$

$(P \lor Q) \to R \Leftrightarrow (P \to R) \lor (Q \to R)$

$P \to (Q \land R) \Leftrightarrow (P \to Q) \land (P \to R)$

$P \to (Q \lor R) \not \Leftrightarrow (P \to Q) \lor (P \to R)$

$P \leftrightarrow (Q \land R) \Leftrightarrow (P \leftrightarrow Q) \land (P \leftrightarrow R)$

$P \leftrightarrow (Q \lor R) \not \Leftrightarrow (P \leftrightarrow Q) \lor (P \leftrightarrow R)$

$P \land (Q \to R) \not \Leftrightarrow (P \land Q) \to (P \land R)$

$P \lor (Q \to R) \Leftrightarrow (P \land Q) \to (P \land R)$

$P \land (Q \leftrightarrow R) \not \Leftrightarrow (P \land Q) \leftrightarrow (P \land R)$

$P \lor (Q \leftrightarrow R) \Leftrightarrow (P \land Q) \leftrightarrow (P \land R)$