newtonian mechanics - Can we do a thought experiment to prove that the horizontal and vertical motion of a projectile are independent of each other?

It seems really hard for me to grasp the idea that horizontal and vertical motion of a projectile are independent of each other. Intuitively I feel that they should affect each other. How can it be that a bullet fired horizontally from the gun and a ball dropped from the same height reach the ground at exactly the same time? Can anyone show me logically why this is true?