Are $[\exists x, \beta \to \alpha(x)]$ and $[\beta \to (\exists x, \alpha(x))]$ the same?

Question

Are $[\exists x, \beta \to \alpha(x)]$ and $[\beta \to (\exists x, \alpha(x))]$ the same?

I think they are the same. Both mean that if $\beta$ is true, then there exists some $x$ for which $\alpha(x)$ is true. Just wanted to confirm.

Answer 1

I am assuming here that when you write $\beta$ instead of $\beta(x)$ that you are implying that the variable $x$ does not occur in $\beta$. That is an important consideration.

With that out of the way, the two statements are not "the same", but they are logically equivalent. The statement on the left is called the prenex-normal form of the statement on the right (assuming that $\alpha$ and $\beta$ are quantifier-free formulas or propositions). An important fact of first-order logic is that all well formed formulas have equivalent formulations that are in prenex-normal form.