Fourier transform and inverse transorm

Question

I have to prove that for $f\in L^1(\mathbb{R})$ $$ \check{\hat{f}}=\hat{\check{f}}, $$ where $$ \hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x $$ and $$\check{f}(\xi):=\frac{1}{2\pi}\int\limits_{\mathbb{R}}e^{i\xi x}f(x)\mathrm{d}x. $$ I know that I have only to apply Fubini's thm and to make a change of variables, but I probably make some errors...