Unsure if two expressions are logically equivalent

Question

I am unsure whether the following statements are logically equivalent.

1. $\forall x((\exists y\text|Q(x,y)) \implies P(x))$

2. $\forall x\forall y(Q(x,y) \implies P(x))$

From my understanding, the first statement should read 'for all X, there exists a $Y$ satisfying $Q(x, y)$ implying $P(x)$'. The second statement can be read as 'for all $X$ and all Y, $Q(x, y)$ implies $P(x)$'.

I did not think these statements were logically equivalent, as they don't seem to convey the same thing. Statement 2 seems to imply statement 1, but statement 1 doesn't necessarily imply statement 2?

I would appreciate any explanation of this above question - does statement 2 also imply statement 1, thereby making the two statements equivalent?

Answer 1

Yes, the two statement are logically equivalent.

In general, we have that

$(\exists v \phi) \to \psi \equiv \forall v(\phi \to \psi)$
$(\forall v \phi) \to \psi \equiv \exists v(\phi \to \psi)$
provided $v$ does not occur free in $\psi$

-- that is, one can move a quantifier between the left-hand side of an implication and outside the implication by switching it to its dual. So $\exists y$ in the left-hand side of $\Longrightarrow$ in statement 1 can become $\forall y$ outside the $\Longrightarrow$ in statement 2 and vice versa.

To get at your intuition, take $Q(x,y)$ to mean "$x > y$", and $P(x)$ for "$x$ is (non-zero) positive". Then "For every natural number $x$, if there is a natural number $y$ that is smaller, then $x$ is positive" should mean the same as "For every pair of natural numbers $x, y$, if the first is greater than the second, then it is positive".