Question 1171515
Alright, let's break down this logical entailment proof. We need to show that from the premise A, ¬F → ¬A, we can derive the conclusion D → (¬E → F).

Here's a step-by-step proof using natural deduction:

**1. A (Premise)**

**2. ¬F → ¬A (Premise)**

**3. ¬¬A (Double Negation Introduction, 1)**

**4. A (Double Negation Elimination, 3)**

**5. F (Modus Ponens, 2, 4)**

**6. ¬E → F (Conditional Introduction, 5)** - Here, we discharge the assumption of ¬E.

**7. D → (¬E → F) (Conditional Introduction, 6)** - Here, we discharge the assumption of D.

**Explanation:**

1.  We start with the given premises, A and ¬F → ¬A.
2.  From A, we can introduce a double negation (¬¬A).
3.  We can eliminate the double negation to get A.
4.  Now, we have A, and ¬F → ¬A. Since A is true, ¬A is false. Therefore, ¬F must be false, which means F is true.
5.  We've derived F.
6.  To prove ¬E → F, we assume ¬E. Since we've already derived F, the implication ¬E → F is true. We discharge the assumption ¬E.
7.  To prove D → (¬E → F), we assume D. Since we've already derived ¬E → F, the implication D → (¬E → F) is true. We discharge the assumption D.

**Therefore, A, ¬F → ¬A ⊢ D → (¬E → F) is a valid entailment.**