Understanding $ \forall x \exists y \forall z P(x, y, z) $

Question

$ \forall x \exists y \forall z P(x, y, z) $

Domain $D$: {1, 2}

For this formula to evaluate to true, would it be correct to say that the predicate $P$ would need to evaluate to true for the following tuples of terms?

$ (1, e_1, 1) $
$ (1, e_1, 2) $

$ (2, e_2, 1) $
$ (2, e_2, 2) $

Where $ e_1 \in D $ and $ e_2 \in D $

Answer 1

Yes, you are right.

You have to show that such $e_1$ and $e_2$ exists to make them all true.