Scott's normal form for $\exists y \forall x R(x,y) $

Question

In this paper Scott's reduction is discussed, which reduces any FO2 formula to Scott's Normal form. As far as I understand the reduction process explained in the paper can get you to formulas with only $\forall \forall$ and $\forall \exists$ quantification prefixes. Which has three possibilities of prefixes $\forall x \forall y $, $\forall x \exists y$ and $\exists y \forall x$. Now scott's normal form only admits $\forall x \forall y $ and $\forall x \exists y$ quantification. Then how is the explained process sound for a formula like $\exists x \forall y R(x,y) $ ?

In other words, what is the scott's reduction for $\Phi(x,y) = \exists x \forall y R(x,y)$.

My attempt on reduction according to the paper:

$ Sc( \Phi(x,y) )= \exists x P(x) \land \forall x \forall y (P(x) \leftrightarrow R(x,y)) $

Answer 1

Your solution is correct. It suffices to see that the formulae $\exists{y} P(y)$ and $\forall{x} \exists{y} P(y)$ are equi-satisfiable, which allows you to put your formulae in the desired form.