Universal Quantifier Distribution

Question

Are the 2 equivalent? Please prove your answer

1) $\forall x(Ax) \to \exists y(By)$

2) $∀x(Ax \to Bx)$

Answer 1

Let $A(t)$ be the assertion that $t$ is divisible by $4$, and let $B(t)$ be the assertion that $t$ is divisible by $3$. Let our domain be the set $\{1,2,3,4\}$ of integers.

The $\forall x A(x)\to \exists yB(y)$ is true, since $\forall xA(x)$ is false. Actually, it is doubly true, since in fact there is a $y$ such that $B(y)$.

But $\forall x(A(x)\to B(x))$ is false.

The two sentences therefore cannot be logically equivalent.

Remark: There is nothing particularly amusing about integers and divisibility. One can undoubtedly give funnier interpretations of $A$ and $B$.