Can someone explain this step in the proof of the convolution theorem?

[codiloo]

I fail to understand a step made in this proof:
http://en.wikipedia.org/wiki/Convolution_theorem"

more specifically the last step where the integral is written as a product of 2 separate integrals (each equal to a Fourier transform):
from:

to:

I'm quite rusty on my integration, but as far I can remember this operation is only allowed when y is independent of x. (since y is taken out of an integral over x). But since we substituted y = z − x this is not the case. Can somebody explain me why this step is correct? (and why I'm wrong)

 

[sfs01]

After the substitution, y is just a dummy variable and consequently independent of x. If it was a definite integral, the substitution would have moved the dependency on x to the integral limits but because the integral is over all space, they are still independent of x after the substitution.