# Does the existence quantifier permute with implication

Let $$A$$ be a statement about $$a$$, and $$B$$ a statement that involves both $$a$$ and $$b$$, (for example $$A(a)$$ is $$a \in \mathbb{R}$$ and $$B(a,b)$$ is $$b\in \mathbb{R} \wedge a . Are the following logical statements equivalent? $$\forall a (A(a) \to \exists b B(a,b))$$ $$\forall a \exists b (A(a) \to B(a,b) )$$ How would one prove this equivalence or if it is not true in general, then what is an example for $$A$$ and $$B$$?