We are tasked with simplifying the expression \[(P \Rightarrow Q) \Rightarrow (\neg Q \Rightarrow \neg P).\]

### Step-by-step simplification:

1. **Recall implications**:
   - The implication \( P \Rightarrow Q \) is logically equivalent to \( \neg P \lor Q \).
   - The implication \( \neg Q \Rightarrow \neg P \) is equivalent to \( P \lor \neg Q \).

   So, we rewrite the expression:
   \[
   (\neg P \lor Q) \Rightarrow (P \lor \neg Q)
   \]

2. **Expand the outer implication**:
   - The implication \( A \Rightarrow B \) is equivalent to \( \neg A \lor B \).
   Applying this to our current expression:
   \[
   \neg (\neg P \lor Q) \lor (P \lor \neg Q)
   \]

3. **Simplify the negation**:
   - Use De Morgan's law to simplify \( \neg (\neg P \lor Q) \):
     \[
     \neg (\neg P \lor Q) = P \land \neg Q
     \]
   So the expression becomes:
   \[
   (P \land \neg Q) \lor (P \lor \neg Q)
   \]

4. **Distribute and simplify**:
   Now, simplify by combining terms:
   - The expression \( (P \land \neg Q) \lor P \) simplifies to just \( P \).
   - The remaining term \( \lor \neg Q \) is already present.
   
   So the final simplified form is:
   \[
   P \lor \neg Q
   \]

Thus, the simplified expression is:
\[
P \lor \neg Q
\]

Would you like further details on any of these steps?

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Here are 5 related questions to deepen your understanding:

1. What is De Morgan's Law, and how does it apply in logic?
2. How do you convert logical implications to disjunctions?
3. Can you simplify \( (P \Rightarrow Q) \land (\neg Q \Rightarrow \neg P) \)?
4. How does \( P \lor \neg Q \) relate to the contrapositive of \( P \Rightarrow Q \)?
5. How do tautologies and contradictions simplify complex logical expressions?

**Tip:** Always remember that an implication \( A \Rightarrow B \) is equivalent to \( \neg A \lor B \), which can greatly help in simplifications.

Learn how to simplify the logical expression (P⟹Q)⟹(¬Q⟹¬P) with a detailed step-by-step solution. This problem explores key concepts in propositional logic, including implications and De Morgan's Law. Suitable for college-level students.

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