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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 \(r\):

\[
\begin{array}{ccc|c|c|c}
p & q & r & \neg p \wedge \neg q & (p \wedge q \wedge r) & [(\neg p \wedge \neg q) \rightarrow (p \wedge q \wedge r)] \iff p \wedge q \\
\hline
T & T & T & F & T & T \iff T \text{ (True)} \\
T & T & F & F & F & T \iff T \text{ (True)} \\
T & F & T & F & F & T \iff F \text{ (False)} \\
T & F & F & F & F & T \iff F \text{ (False)} \\
F & T & T & F & F & T \iff F \text{ (False)} \\
F & T & F & F & F & T \iff F \text{ (False)} \\
F & F & T & T & F & F \iff F \text{ (True)} \\
F & F & F & T & F & F \iff F \text{ (True)} \\
\end{array}
\]

From the truth table, we see that the expression is true in some cases and false in others. It's evident that the bi-conditional statement does not hold for all cases; thus, the expression is not a tautology.

Would you like any further explanations on any of the steps? Here are some follow-up questions and a tip to consider:

1. How does the implication logical operator work in more detail?
2. What happens if we change the conjunction on the right-hand side of the implication to a disjunction?
3. Can you provide real-life examples of when such logical expressions might be used?
4. What is the impact of including more variables in such logical expressions?
5. How do different values of \(r\) affect the overall truth value of the expression?

**Tip**: Exploring variations of complex logical expressions by altering logical operators or the variables involved can offer deeper insights into logical structures and their applications.

[(¬p ∧ ¬q) → (p ∧ q ∧ r)] ⟷ p ∧ q

Solve and understand the logical equivalence problem involving conjunctions, negations, implications, and bi-conditionals. Detailed analysis with truth tables provided to evaluate the expression [(¬p ∧ ¬q) → (p ∧ q ∧ r)] ⟷ p ∧ q. Suitable for high school or introductory college-level logic students.

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