Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 1

Corollary to Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation
Consider the Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation. Let $\dfrac {\partial \Phi} {\partial x} = 0$.

Then $\Phi$ is the first integral if its Poisson Bracket with the Hamiltonian vanishes.

Proof
Set $\dfrac {\partial \Phi} {\partial x} = 0$ in Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation.