We study metric transformations which depend on a scalar field $phi$ and its first derivatives and confirm that the number of physical degrees of freedom does not change under such transformations, as long as they are not singular. We perform a Hamiltonian analysis of a simple model in the gauge $phi = t$. In addition, we explicitly show that the transformation and the gauge fixing do commute in transforming the action. We then extend the analysis to more general gravitational theories and transformations in general gauges. We verify that the set of all constraints and the constraint algebra are left unchanged by such transformations and conclude that the number of degrees of freedom is not modified by a regular and invertible generic transformation among two metrics. We also discuss the implications on the recently called hidden constraints and on the case of a singular transformation, a.k.a. mimetic gravity.
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