Question 1171517
Let's break down this entailment step-by-step.

**Given:**

* α and β are logically equivalent. This means they have the same truth values under every valuation (interpretation) of their sentence letters.

**Entailment to Prove:**

* β → α, α ∨ β ⊨ α ∧ β

**Analysis:**

1.  **Logical Equivalence:**
    * Since α and β are logically equivalent, we can say α ≡ β. This means that whenever α is true, β is true, and whenever α is false, β is false.

2.  **β → α:**
    * Because α ≡ β, the implication β → α is always true. If β is true, α is true, and if β is false, α is false. Thus the implication is always true.

3.  **α ∨ β:**
    * Since α ≡ β, α ∨ β will be true whenever either α or β (or both) are true. In fact, due to logical equivalence, α ∨ β is true when α is true, and also true when β is true.

4.  **α ∧ β:**
    * We want to show that α ∧ β is true in all cases where β → α and α ∨ β are true.
    * Since β → α is always true, we only need to consider α ∨ β.
    * If α ∨ β is true, then at least one of α or β is true.
    * Because α ≡ β, if one is true, the other is also true.
    * Therefore, both α and β are true.
    * Hence, α ∧ β is true.

5.  **Entailment:**
    * We need to check if in every valuation where β → α and α ∨ β are true, α ∧ β is also true.
    * β → α is always true.
    * If α ∨ β is true, then both α and β are true (due to logical equivalence).
    * Therefore, α ∧ β is true.
    * Thus, the entailment holds.

**Conclusion:**

Yes, the entailment β → α, α ∨ β ⊨ α ∧ β holds.