Fix two concept names A and B as well as some role name r. Does the entailment {∀r.A ⊆ ∃r.A} |= ∀r.B ⊆ ∃r.B hold true?

Yes, the entailment {∀r.A ⊆ ∃r.A} |= ∀r.B ⊆ ∃r.B holds true.

This can be proven using the principle of subsumption, which states that if A ⊆ B, then ∀r.A ⊆ ∀r.B and ∃r.B ⊆ ∃r.A. Since A ⊆ A, we have ∀r.A ⊆ ∃r.A.

Therefore, by applying the principle of subsumption, we have ∀r.B ⊆ ∀r.A and ∃r.A ⊆ ∃r.B. Combining these two inclusions, we get:

∀r.B ⊆ ∃r.B

Thus, the entailment holds true.