I need convert $ \forall x \forall y(P(x,y) \sim Q(x,y)) \vee \exists x \exists y(P(x,y) \sim Q(y))$ into Prenex Normal Form.
Can I use this formula: $(\exists x\phi )\lor \psi $ is equivalent to $\exists x(\phi \lor \psi )$ like $$ \forall x \forall y(P(x,y) \sim Q(x,y)) \vee \exists x \exists y(P(x,y) \sim Q(y)) $$ is equivalent to $$ \exists x \forall x \forall y(P(x,y) \sim Q(x,y)) \vee \exists y(P(x,y) \sim Q(y))$$ is equivalent to $$ \exists y \exists x \forall x \forall y(P(x,y) \sim Q(x,y)) \vee (P(x,y) \sim Q(y))$$ Is it converting to Prenex Normal Form?