Wrong. Homo Sapien Sapien evolved from Homo Sapien. Neither evolved from Neandertal, according to DNA evidence. Unless somebody comes up with evidence to counter it, it stands. Neandertals are extinct, for whatever reason.
While the two species existed at the same time, Neanderthal did not evolve into Homo sapiens. This is well documented by DNA evidence.
You obviously need to study the theory of evolution a little more before you can adequately dispute it. Right now you are just looking silly.
The following chart shows what one expert thinks of the process of hominid evolution:
Source: http://wwwrses.anu.edu.au/environment/eePages/eeDating/HumanEvol_info.html
Why are you cackling like a loon for no particular reason. Are you somehow childishly amused by a standard taxonomic designation? (Okay, who let all the four-year-olds in?)
Ok, then I guess a better example would be from Neanderthal to Cromagnon then?
No, it wouldn't.
Except that theories purport that these two species existed at the same time.
It's not the "theories" that "purport" this, you goof, it's an established fact.
Not necessarily evidence that one didn't evolve from the other, though, right?
Right, actually, but I'll bet you can't figure out that one either without a great deal of assistance. (Hint: My mother and I are both alive at the same time, does that mean she can't possibly be my ancestor?)
In this particular case, though, the evidence is extremely strong that both species of human evolved from a common ancestor, and not one from the other.