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The given logical expression appears to involve rules of inference in propositional logic. Let's break it down and solve it using these rules.

### Expression:
1. \( (a \land b) \to [a \to (d \lor e)] \)
2. \( (a \land b) \land c \)
3. Therefore: \( d \lor e \)

### Steps to solve:

1. From line 2: \( (a \land b) \land c \), we can apply **simplification** to extract \( a \land b \).
   
   \[
   (a \land b) \land c \implies a \land b
   \]

2. From the first premise: \( (a \land b) \to [a \to (d \lor e)] \), and using **modus ponens** (if \( p \to q \) and \( p \), then \( q \)) with the result \( a \land b \), we can conclude:

   \[
   a \to (d \lor e)
   \]

3. Now, we know \( a \to (d \lor e) \). To derive \( d \lor e \), we need to affirm \( a \). 

   From \( a \land b \), we can apply **simplification** again to extract \( a \):

   \[
   a \land b \implies a
   \]

4. Finally, applying **modus ponens** again to \( a \to (d \lor e) \) with \( a \), we conclude:

   \[
   d \lor e
   \]

Thus, using the rule of inference, we can derive that \( d \lor e \) holds.

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Would you like further details or clarifications?

### Related Questions:
1. What are the basic rules of inference in propositional logic?
2. How does **modus ponens** work in logical derivations?
3. What is the **simplification** rule, and when is it applied?
4. Can we use **disjunction elimination** in similar problems?
5. How do we prove logical equivalences using truth tables?

### Tip:
When solving problems involving logical rules, focus on identifying key propositions that allow you to apply **modus ponens** or **simplification**, as these often lead to quick derivations.

Use the rule of inference to solve the following expression: (a ∧ b) → [a → (d ∨ e)], (a ∧ b) ∧ c, therefore: d ∨ e.

Learn how to apply the rules of inference to solve the logical expression: (a ∧ b) → [a → (d ∨ e)], (a ∧ b) ∧ c, therefore: d ∨ e. This solution involves using Modus Ponens and the Simplification Rule, key concepts in propositional logic.

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