To simplify the expression \(\left((P \Rightarrow Q) \Rightarrow P\right) \Rightarrow Q\), we can break it down using logical equivalences step by step.

1. **Rewrite implications in terms of disjunctions:**
   The implication \(A \Rightarrow B\) is logically equivalent to \(\neg A \vee B\). So we can rewrite the expression as:

   \[
   \left((\neg P \vee Q) \Rightarrow P\right) \Rightarrow Q
   \]

2. **Apply the implication equivalence to \((\neg P \vee Q) \Rightarrow P\):**
   Now rewrite the inner implication \((\neg P \vee Q) \Rightarrow P\):

   \[
   \neg(\neg P \vee Q) \vee P
   \]

   By De Morgan's Law, \(\neg(\neg P \vee Q)\) is equivalent to \(\neg \neg P \wedge \neg Q\), which simplifies to \(P \wedge \neg Q\). So, the expression becomes:

   \[
   (P \wedge \neg Q) \vee P
   \]

3. **Simplify \((P \wedge \neg Q) \vee P\):**
   Using the distributive property, \((P \wedge \neg Q) \vee P\) simplifies to just \(P\), because \(P \vee (P \wedge \neg Q) = P\) (since \(P\) is already included in both terms).

   So, the expression simplifies to:

   \[
   P
   \]

4. **Substitute back into the original expression:**
   Now we substitute \(P\) into the original outer implication:

   \[
   P \Rightarrow Q
   \]

5. **Final simplification:**
   The expression \(P \Rightarrow Q\) is already in its simplest form, so the final simplified form of the entire expression is:

   \[
   P \Rightarrow Q
   \]

### Final Answer:
\[
(P \Rightarrow Q) \Rightarrow P \Rightarrow Q \text{ simplifies to } P \Rightarrow Q.
\]

Would you like more details or have any questions? Here are some related questions:

1. What are some other common logical equivalences used in simplification?
2. How can De Morgan's Laws help in simplifying logical expressions?
3. Can the expression \(P \Rightarrow Q\) be expanded using different methods?
4. What role does distributive property play in simplifying logical expressions?
5. How does simplification impact the truth table of a logical statement?

**Tip:** When simplifying logical expressions, converting implications to disjunctions and using De Morgan's laws can greatly simplify your work.

Learn how to simplify the logical expression ((P ⟹ Q) ⟹ P) ⟹ Q step-by-step using logical equivalences such as implications, De Morgan's Law, and distributive properties. This solution is perfect for university-level logic students or advanced high school math learners.

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