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The logical expression you have uploaded appears to involve conditional (→), biconditional (↔), negation (∼), and conjunction (∧) operators. Let's break it down step by step for clarity:

The expression is:

\[
\left[ (p \rightarrow q) \rightarrow r \right] \leftrightarrow \sim \left[ (\sim q \rightarrow \sim p) \land \sim r \right]
\]

### Breaking it down:
1. **Left Side**:
   - \((p \rightarrow q)\) represents the implication "if \(p\), then \(q\)".
   - \((p \rightarrow q) \rightarrow r\) says "if \(p \rightarrow q\), then \(r\)".
   
2. **Right Side**:
   - \(\sim q \rightarrow \sim p\) is the contrapositive of \(p \rightarrow q\), which says "if not \(q\), then not \(p\)".
   - \((\sim q \rightarrow \sim p) \land \sim r\) combines the contrapositive with \(\sim r\) using a conjunction (AND).
   - \(\sim [(\sim q \rightarrow \sim p) \land \sim r]\) negates the entire right side.

### Biconditional (\(\leftrightarrow\)):
The entire expression asserts the equivalence (biconditional) between the left side \(((p \rightarrow q) \rightarrow r)\) and the negation of the right side \(\sim [(\sim q \rightarrow \sim p) \land \sim r]\).

### Next Steps:
This expression can be simplified or evaluated using logical equivalences like De Morgan's laws, contrapositive, and the properties of implication. Would you like to proceed with simplifying this further or applying specific logical rules to analyze it? 

Feel free to ask for more details or further clarifications!

Here are 5 related questions you might find useful:
1. Would you like a step-by-step simplification of the logical expression?
2. Should I evaluate the truth table for this expression?
3. Are you interested in logical equivalence rules such as De Morgan’s laws?
4. Do you want to verify the validity of this biconditional in formal logic?
5. Would you like to explore how this expression applies in propositional logic?

**Tip**: Remember that in logical proofs, simplifying implications using contrapositive or De Morgan's laws can make complex expressions more manageable.

[(p → q) → r] ↔ ~ [(~ q → ~ p) ∧ ~ r]

Explore the logical breakdown of the complex expression [(p → q) → r] ↔ ~ [(~ q → ~ p) ∧ ~ r]. This problem covers propositional logic, including implications, negations, conjunctions, and the biconditional operator, with applications of De Morgan’s laws and contrapositive transformations. Suitable for advanced logic students.

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