I'm a little confused with logical equivalences of quantifications. Particularly, universal. I know that ∀x(P(x)∨Q(x)) is not logically equivalent to ∀xP(x) ∨ ∀xQ(x), but I don't know the proper way to show it.
But, what I am more concerned with is if ∀x[P(x)↔Q(x)] and ∀xP(x) ↔ ∀xq(x) are logically equivalent. I would like to understand this. Thanks