= Solution
Set $D=\partial_t+\{\,\cdot\,,H\}$. The hypotheses mean $Df=Dg=0$ as identities on <phase space>. By the <Jacobi identity for the Poisson bracket>,
$$
\{\{f,g\},H\}=\{\{f,H\},g\}+\{f,\{g,H\}\}.
$$
Also, the <product rule> applied to the defining coordinate formula gives
$$
\partial_t\{f,g\}=\{f_t,g\}+\{f,g_t\}.
$$
Adding these identities proves
$$
D\{f,g\}=\{Df,g\}+\{f,Dg\}.
$$
Consequently
$$
\boxed{\frac{d}{dt}\{f,g\}=0.}
$$
This proves the <Poisson bracket of time-dependent conserved quantities> theorem, including explicit time dependence of $f$ and $g$.
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