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 $b\in \mathbb{R} \wedge a<b)$ . 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 $B$?


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