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.