Is S^{c} the key difference in this logical equivalence?

[Horse]

Is it equivalent?
$$( \forall x \in S \forall y P(x) ) <=> \neg ( \exists x \in S^{c} \exists y \neg P(x) )$$

Attempt at solution
I think it should be
$$( \forall x \in S \forall y P(x) ) <=> \neg ( \exists x \in S \exists y \neg P(x) )$$
The diiference to the above statement is $$S^{c}$$.

 

[HallsofIvy]

Yes, your statement is correct.

 

[Horse]

Yes, your statement is correct.

Please, can you clarify whether it is the first statement or my attempt at solution?