I don't know that this solution is right

Question

I have the statement ∃xP(x) ↔ ∃xQ(x) and I have to write it in prenex normal form.

First I rewrote it as two conditionals:

∃xP(x) ↔ ∃xQ(x) ≡(∃xP(x) → ∃xQ(x))∧ (∃xQ(x) → ∃xP(x))

Then I renamed the variables, so they all have a unique name.

≡(∃xP(x) → ∃yQ(y))∧ (∃zQ(z) → ∃wP(w))

Next, I pull out the quantifiers.

≡∃x∃y(P(x) → Q(y))∧ ∃z∃w(Q(z) → P(w))

≡∃x∃y∃z∃w(P(x) → Q(y))∧ (Q(z) → P(w))

Is that right?