True or False: $\exists x(P(x)\lor Q(x))\equiv \exists xP(x)\lor \exists yQ(y)$
My intuition tells me yes, these two things are equivalent. Assume the first, take some $x_0$ s.t. $P(x)\lor Q(x)$, do a proof by cases that the second is true as well. And similarly the other way.
But the proof seems really fast, and kind of trivial. Maybe that's just because the question itself is trivial, or maybe my intuition is incorrect!
First, are those two statements indeed equivalent? Secondly, is my idea for the proof headed in the right direction?