Delta-distribution composed with a function from the Fourier representation

A well known representation of Dirac’s delta-distribution is via the Fourier transform of distributions: \begin{equation} \delta[f]:=f(0)=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}xk}f(k)\mathrm{d}k\mathrm{d}x. \end{equation} Can this be used to define the delta distribution composed with a function $ \phi:\mathbb{R}\to\mathbb{R}$ via \begin{equation} (\delta\circ\phi)[f]:=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}x\phi(k)}f(k)\mathrm{d}k\mathrm{d}x? \end{equation} Does this make sense? Heuristically and in a more physics-style notation, I would argue that \begin{equation} \int\int e^{\mathrm{i}x\phi(k)}f(k)\mathrm{d}k\mathrm{d}x “=” \int\left(\int e^{\mathrm{i}x\phi(k)}\mathrm{d}x\right)f(k)\mathrm{d}k “=” \int\delta(\phi(k))f(k)\mathrm{d}k. \end{equation} If it does not make sense, what can I say about the above expression for general functions $ \phi$ ? E.g. I would expect the above integral to be positive if $ f$ is point-wise positive.