All 3 corollaries are valid. I think the reason why it is not mentioned even is the fact of its limited applicability compared to the maximum entropy principle. Most books of physical chemistry don't even mention the energy minimum principle. Suppose that a system undergoes a change at constant U like the isotermal expansion of a ideal gas inside an isolated vessel at constant V. How can You establish if the system has attained the equilibrium using the energy/volume corollaries? U and V are at a minimum since the beginning of the expansion... Same thing for all isochoric/isoenergetic transformations. It is the entropy the true indicator that allows to follow the evolution of a system toward equilibrium.
The proof of the equivalence of the two extremum criteria for S and U can be formulated either as a physical or mathematical argument.
- $dS_{U,V , n_i} = 0$
- $d^2 S_{U,V, n_i} \le 0$
the first equation says that the first derivative vanishes at an extremum, whereas the second one says that, if the second derivative is negative at the extremum, then the extremum is a maximum.
it can be shown that
\begin{equation}
\left(\frac{\partial U} {\partial X} \right)_{S}= -T \left(\frac{\partial U} {\partial X} \right)_{U}
\end{equation}
if $(\partial S/\partial X)_U = 0$ vanishes, the so does $(\partial U/\partial X)_U = S$
where $X$ can be any of the variable $V,n_i$
Thus an extremeum for entropy at constant internal energy is also an extremum for internal energy at constant entropy and by calculating the second derivative one finds that:
\begin{equation}
\left(\frac{\partial U} {\partial X} \right)_{S}= -T \left(\frac{\partial^2 S} {\partial X^2} \right)_{U} > 0
\end{equation}
The Second Law of Thermodynamics therefore leads to a minimum energy principle: at equilibrium in a constant entropy system the internal energy takes it minimum possible value. In the notation of thermodynamics we write the minimum energy as
$dU_{S,V , n_i} =0$
$d^2 U_{S,V,n_i} \ge 0$
from a mathematical perspective considering now the volume the situation is analogous as can be seen by the correspondence of the differentials of U and V
$dU = -P dV + T dS$
$dV = -\frac{1}{P} dU +\frac{T}{P} dS$
we could graph both functions for the minimum (here shown for U) as

The equilibrium state A as a point of minimum U for constant S.
And write
$dV_{S,U , n_i} =0$
$d^2V_{S,U,n_i} \ge 0$
Now I will show the consistency of the energy minimum principle and of the minimum volume principle. Starting from the energy minimum principle; But I want to make a consideration first.
The entropy maximum and the energy minimum principles are both valid definitions of thermodynamic equilibrium. It is necessary to be extremely careful here, though. The equilibrium state defined by equations (3) and (4) and (5) (6) is not the same as the one defined by equations (1) and (2). It cannot possibly be, as in the first case we are dealing with an isolated system and in the second case we are not. If entropy were kept
constant in an isolated system then the system would not change and the minimum energy principle would be meaningless, because internal energy would not be able to change. In order for constant entropy minimization of the internal energy to be possible, the system of interest must be part of a larger isolated system in which entropy does increase.
Consider the example of the two bodies forming the system A at different temperatures, $T_2 > T_1$, shown now in the Fig. below; we want to achieve thermal equilibrium at constant entropy and volume, then system A cannot be isolated. It must be able to exchange energy with its surroundings, which conform an isolated system, labeled B in the figure, and the entropy of B must increase. In order for the entropy of B to increase there must be heat transfer between A and the rest of B. Because equilibrium of A at constant entropy requires that its internal energy be minimized, heat must be transferred from A to its surroundings. Therefore, the final temperature of A if it is allowed to reach equilibrium at constant entropy must be lower than its final temperature if it reaches equilibrium at constant internal energy.

To show the consistency of the volume minimum principle from a physical perspective, consider the same system A now isoentropic and also isoenergetic and also compressible, initially not in mechanical equilibrium ($P^A < P^{\text{ext}}$) since it has not reached the minimum possible volume. In order to attain the equilibrium work should be done on the system compressing it and heat should be transferred from A to its surroundings to keep the system at S and U constant;