Equivalency of Two First-Order Formulae I was reading a paper, and I encountered a definition of some concept. The definition was of the form: (+) $\ \qquad\qquad (\exists x) \phi(x) \quad \Rightarrow \quad (\exists y) \psi(y)$ where $\phi$ and $\psi$ are two formulae. I was wondering if the author's could write the above formula in the form: (++) $\qquad\qquad (\forall x)(\exists y) \quad \phi(x) \Rightarrow \psi(y)$ That is, if (+) and (++) are equivalent. Please prove the equivalency, or give a counterexample.

Yes the two formulas say the same. $((\exists x)\phi(x))\to((\exists y)\psi(y))$ is equivalent to $(\lnot(\exists x)\phi(x))\lor((\exists y)\psi(y))$ which is equivalent to $((\forall x)\lnot\phi(x))\lor((\exists y)\psi(y))$ which is equivalent to $(\forall x)(\exists y)(\lnot\phi(x)\lor\psi(y))$ which is what you are looking for, namely $(\forall x)(\exists y)(\phi(x)\to\psi(y))$.