The equivalence of the three formulations of the second law that concern themselves with heat engines and heat pumps is usually established by a series of demonstrations that if you violate one of them you can violate the others.
Do all the combinations and you have established that each is necessary and sufficient for the others.
Example. Assume I have a machine that violates the Clausius statement. If I combine it with an ordinary heat engine, I can move the waste heat of the engine back to the hot reservoir, so that the combined machine is converting all of it's net heat drawn from the hot reservoir to work: a violation of the Kelvin-Plank statement.
Working through all the combinations is tedious but not hugely difficult.