We are tasked with determining whether the logical equivalence below holds:

\[
p \to (q \to r) \leftrightarrow (p \land q) \to r
\]

### Step 1: Simplify each side of the biconditional statement.

#### Left-hand side: \( p \to (q \to r) \)
This expression represents an implication where \( p \) implies another implication, \( q \to r \).

We can rewrite \( q \to r \) as \( \neg q \lor r \). Therefore, the entire left-hand side becomes:
\[
p \to (\neg q \lor r)
\]
Using the material implication rule \( p \to A \equiv \neg p \lor A \), we rewrite \( p \to (\neg q \lor r) \) as:
\[
\neg p \lor (\neg q \lor r)
\]
This simplifies to:
\[
\neg p \lor \neg q \lor r
\]

#### Right-hand side: \( (p \land q) \to r \)
Again using the material implication rule, we rewrite \( (p \land q) \to r \) as:
\[
\neg (p \land q) \lor r
\]
We know that \( \neg (p \land q) \) is logically equivalent to \( \neg p \lor \neg q \), so this becomes:
\[
(\neg p \lor \neg q) \lor r
\]
which simplifies to:
\[
\neg p \lor \neg q \lor r
\]

### Step 2: Compare the two sides

The left-hand side simplifies to \( \neg p \lor \neg q \lor r \), and the right-hand side also simplifies to \( \neg p \lor \neg q \lor r \).

Since both sides are equivalent, we conclude that the original statement is true:
\[
p \to (q \to r) \leftrightarrow (p \land q) \to r
\]

### Conclusion
The given statement is a tautology, meaning it holds true for all truth values of \( p \), \( q \), and \( r \).

Would you like more details on the steps or have any further questions?

Here are some related questions for further practice:

1. What is the truth table for \( p \to (q \to r) \)?
2. How can De Morgan's laws help simplify logical expressions?
3. How is the material implication rule applied in propositional logic?
4. What is the contrapositive of \( p \to (q \to r) \)?
5. Can you prove this equivalence using truth tables instead of symbolic manipulation?

**Tip:** Always check if both sides of a logical equivalence simplify to the same form, as it can make the verification much easier!

Explore the logical equivalence between p → (q → r) and (p ∧ q) → r with a detailed, step-by-step breakdown. Learn key concepts like material implication and biconditional logic, and discover how both sides simplify to the same form using propositional logic. Suitable for students in undergraduate logic or discrete mathematics courses.

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