I know that enthalpy is defined as $H=E+PV$. So for a system the enthalpy is the sum of internal energy and the amount of work needed at constant pressure $P$ to give it's volume $V$. But the differential of enthalpy is given by legendre transformation, $$dH=TdS+VdP$$ For an adiabatic system ($dQ=0$) if work is done on the system at a constant pressure the mechanical equilibrium is achived when enthalpy is minimized. So whenever there is some work done on the system at constant pressure with no heat exchange, $dH$ is negative.
So my question is, if $dQ=0$, then $TdS$ is also zero and at constant pressure $dP=0$. So by the definion of enthalpy, $dH$ is not zero it's negative, but from the differential $dH=0$ always. Can anyone explain how do I interpret this?