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

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SUMMARY

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, each representing a Fourier transform. The confusion arises from the substitution of variables, specifically the transition from y to z - x. It is established that after this substitution, y acts as a dummy variable, making it independent of x, which justifies the separation of the integrals. This clarification is crucial for understanding the validity of the convolution theorem's proof.

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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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