[QUE/CM-04004]

$\newcommand{\PB}[1]{\{#1\}_\text{PB}}$ \(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)
Let $\Psi$ be a function of $f,g,h,..$ which are in turn functions of  $q$ and $p$. Let $ \phi({{\bf q}, {\bf p}})$ be a function of \(\bf q,p\),  Show that the Poisson bracket of $\phi$ with $\Psi$ is given by $$\PB{\phi,\Psi} =\pp[\Psi]{f}\PB{\phi,f} + \pp[\Psi]{g}\PB{\phi,g}  + \pp[\Psi]{h}\PB{\phi,h} + \cdots $$