Second consequence of invariance under regular canonical transformation in Shankar's QM book

Near the end of chapter 2 of R. Shankar’s book Principles of Quantum Mechanics, he talks about two consequences of invariance of the Hamiltonian under a regular canonical transformation. My problem is with the second consequence and it’s proof (pg 99 and 103 respectively) :

If $H$ is invariant under the regular canonical, but not necessarily infinitesimal, transformation $(q,p) to (bar{q}, bar{p})$ and if $(q(t),p(t))$is a solution to the equations of motion, so is the transformed (translated, rotated, etc) trajectory $(bar{q}(t), bar{p}(t))$.

In his proof of this consequence he tries to show that $bar{q}$ and $bar{p}$ satisfy
$$dot{bar{q}} = frac{partial H}{partial bar{p}}$$
$$dot{bar{p}} = – frac{partial H}{partial bar{q}}$$
using the invariance of the Hamiltonian
$$H(q,p) = H(bar{q}, bar{p}).$$
But why should the invariance of the Hamiltonian matter at all? If the transformation $(q,p) to (bar{q}, bar{p})$ is canonical then by definition of canonical transformations, $bar{q}$ and $bar{p}$ should already satisfy Hamilton’s equations. Shouldn’t this be a consequence of just canonical transformations in general rather than being a consequence of the invariance of the Hamiltonian?