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The change in enthalpy isn't the change in heat without work, it refers to the transfer of heat at a constant pressure, which implies that work is being done. Enthalpy is the total energy content of a system so the change in enthalpy is equal to the change in internal energy due to transfer of heat plus work (PV) because work is being done on the system so that it stays at the constant pressure. Remember that U = H - PV and PV is negative because work is being done by the system. That equation is also just another form of your equation.
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