How to prove that predicates are not equivalent?

Question

Given following predicates:

$$ F_1 = (\forall x)(F(x) \leftrightarrow G(x)) \text{ and } F_2 = (\forall x)F(x) \leftrightarrow (\forall x)G(x) $$

I think that they are not equivalent, but if it possible to prove that?

Answer 1

Consider a domain with exactly two objects, one of which has property $F$ but not property $G$, while the other one has property $G$, but not property $F$.

Then $(\forall x)(F(x) \leftrightarrow G(x))$ is clearly false, but since $(\forall x)F(x)$ and $(\forall x)G(x)$ are both false as well, $(\forall x)F(x) \leftrightarrow (\forall x)G(x)$ ends up being true

Answer 2

Consider $a,b\in\mathbb R$. $$\forall x(ax + b = 0) \leftrightarrow \forall x (a = b = 0)$$ certainly holds, but $$\forall x (ax + b = 0 \leftrightarrow a = b = 0)$$ does not!