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

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The discussion centers on a specific step in the proof of the convolution theorem, particularly the transition from an integral to a product of two separate integrals representing Fourier transforms. The confusion arises from the substitution of variables, where y is initially dependent on x. However, it is clarified that after the substitution, y becomes a dummy variable, rendering it independent of x. This independence allows the separation of the integrals despite the initial dependency. The key takeaway is that in the context of an integral over all space, the variables can be treated as independent after substitution.
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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:
30aa5ce6a4881f46515121f4ccfee81d.png

to:
97cc52195eb954a68ef235c3969e3f02.png

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)
 
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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.
 

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