Find Prenex form and skolem form

Question

We have a formula:

$(\exists x)R(x,y)\iff(\forall y)P(x,y)$

Find a prenex form of the formula and convert it into skolem variant.

My solution

Is it correct?

Answer 1

Since on step 1 you have $\neg \forall y P(x,y)$, you get $\exists y \neg P(x,y)$, and thus you end up with $\exists d$ rather than $\forall d$ on line 2.

Also, I would move this $\exists d$ right after the $\exists a$, otherwise you would need to use functions when doing skolemization, and fortunately you can do that by first bringing out the $a$ and $d$ using the Prenex laws before bringing out the $b$ and $c$. (Also, if you get graded on your work, I would show all those steps)

And so on the last step you just need a constant instead of the $d$ (And of course you get rid of that quantifier)