I've been trying to understand the time shift proof with fourier transforms, however I'm confused as to how some variables can change but not others. My intuitive understanding of the time shift is as follows:
We define $f(y) = f(x-c)$
$$ \mathcal{F}[f(x-c)] (s) = \int^{\infty}_{-\infty} f(y) e^{-isy} dy $$ $$ = \int^{\infty}_{-\infty} f(x-c) e^{-isx} dx $$ The confusion comes when it's chosen that $x = x+c$, and $f(x-c) = f(x)$. I dont't understand why this can be done. Has this something to do with the fact that the function is integrated across the whole real line? Since $y = x-c$, why wouldn't $e^{-isy} = e^{-is(x-c)}$?
I looked at this similar question, however it provided little clarification on why we $e^{-isy}$ to $e^{-is(x+c)}$.