The different disciplines of science may be able to be used to supplement and describe properties of other disciplines. The different disciplines often work together to define more than any one would be able to do by itself. That does not mean that the different disciplines should be treated as if they are completely interchangeable as is being done with the first law of thermodynamics.
Nobody is doing that. What has been discovered is that there is a equivalence between mass and energy. All that means is that if a certain amount of mass is converted into energy, the system will remain in balance because the energy that is created equals the mass that was used to create it. If energy is converted into mass then the amount of mass created will offset the loss of energy exactly. The laws of conservation of mass and the conservation of energy are not violated. This is what happens in science. As we learn more things, we sometimes have to reexamine things. In this case, E=mc^2 and its relationship to the laws of thermodynamics needed to be reexamined and it was found that E=mc^2 doesn't change anything. The implications become somewhat broader, but that is all. In all honesty, E=mc^2 has virtually no impact on thermodynamics because unless mass is converted to energy or energy is converted to mass, who cares. If we are going to convert mass into energy, well, lets just say that thermodynamics is not generally used to design nuclear weapons.