How do I show if (∀x∃y∀zP(x, y, z)) ⇒ (∃y∀x∀zP(x, y, z)) is a logically valid statement?

Question

I have tried to add a double negation to the left hand side and then swap the existential quantifiers, which gives:

¬¬∀x ∃y ∀z P(x, y, z) ⇔

¬∃x ¬∃y ∀z P(x, y, z)

This, I thought, means I can swap the x and y around because you can swap two of the same quantifiers around. This gives:

¬∃y ¬∃x ∀z P(x, y, z)

but when pushing the negation back to the left I get:

∀y ∃x ∀z P(x, y, z)

which means that ∀x ∃y ⇔ ∀y ∃x. Is this true, or even helpful to solving the question?

Answer 1

Take $P(x,y,z) = x<y$ for a simple counterexample.