We cannot do the reverse of this though. If there is some \(y\) for which every \(x\) satisfies \(P(x,y)\text{,}\) then certainly for every \(x\) there is some \(y\) which satisfies \(P(x,y)\text{.}\) The first is saying we can find one \(y\) that works for every \(x\text{.}\) The second allows different \(y\)'s to work for different \(x\)'s, but there is nothing preventing us from using the same \(y\) that work for every \(x\text{.}\) In other words, while we don't have logical equivalence between the two statements, we do have a valid deduction rule:
in-context