The expression you provided involves logical propositions and can be analyzed using truth tables or logical equivalences. The expression given is:

$[(\neg p \wedge \neg q) \rightarrow (p \wedge q \wedge r)] \iff p \wedge q$

Let's break down this expression step by step:

1. Negation and Conjunction:

   • $\neg p \wedge \neg q$: This expression is true only when both $p$ and $q$ are false.

2. Implication:

   • An implication $(A \rightarrow B)$ is false only if $A$ is true and $B$ is false. Otherwise, it is true.

3. Right side of Implication:

   • $p \wedge q \wedge r$: This is true only when $p$, $q$, and $r$ are all true.

Given the first part $\neg p \wedge \neg q$ is true when both $p$ and $q$ are false, the implication $(p \wedge q \wedge r)$ will be vacuously true (since $\neg p \wedge \neg q$ being true guarantees that $p \wedge q \wedge r$ is false). Therefore, the implication will always be true except in the scenario where $p$ and $q$ are true but $r$ is false.

4. Bi-conditional:
   • $(A \iff B)$ is true if both $A$ and $B$ are either true or false together.

To determine the truth value of the entire expression, we need to evaluate the bi-conditional relation to see if the left side of the implication is logically equivalent to $p \wedge q$.

Now, let's build a truth table to examine the truth values for all combinations of $p$, $q$, and $r$: