Question about Fouriers Inversion Theorem. This is my professors definition of the FIT.

Fourier Inversion Theorem: Assume that $f\in L^2(\mathbb{R}).$ Define the Fourier transform to be
  $$\widehat{f}(\xi)=\int\limits_{\mathbb{R}}f(y)e^{-iy\xi} \ dy.\tag 1$$
  Then, as an equality in $L^2(\mathbb{R})$ we have the inverse
$$f(x)=\frac{1}{2\pi}\int\limits_{\mathbb{R}}\widehat{f}(y)e^{ixy} \ dy. \tag2$$

These notations confuse me a lot. In $(2)$, don't we actually want to get back $f(y)$ from $\widehat{f}(t)$ just like it says here? But then Wikipedia kind of contradicts(?) itself here.
Can anyone explain what is going on here with the symbols? I understand it as if we start with $f(y)$ and apply the tranform, we get $f(\xi).$ So far so good. But when we apply the inverse transform, It should be applied to $\widehat{f}(\xi)$ and give us back our original $f(y)$?

The choice of variables that were used is confusing. Note the $x$ and $y$ in your second equation of
$$f(x)=\frac{1}{2\pi}\int\limits_{\mathbb{R}}\widehat{f}(y)e^{ixy} \ dy. \tag2$$
are basically "dummy" variables, with $x$ being a placeholder for the variable of the function $f$ and $y$ specifying the variable being integrated. As such, it's just as accurate to use
$$f(y)=\frac{1}{2\pi}\int\limits_{\mathbb{R}}\widehat{f}(\xi)e^{iy\xi} \ d\xi. \tag3$$
instead, where I've replaced the $y$ with $\xi$ and $x$ with $y$. This shows the inverse transform does use $\widehat{f}(\xi)$ to go back to your original $f(y)$. I trust this help to explain the issue to you.
