We use the interactive theorem prover Isabelle to prove that the algebraic axiomatization of bisimulation

equivalence in the pi-calculus is sound and complete. This is the first proof of its kind to be wholly machine

checked. Although the result has been known for some time the proof had parts which needed careful

attention to detail to become completely formal. It is not that the result was ever in doubt; rather, our

contribution lies in the methodology to prove completeness and get absolute certainty that the proof is

correct, while at the same time following the intuitive lines of reasoning of the original proof. Completeness

of axiomatizations is relevant for many variants of the calculus, so our method has applications beyond

this single result. We build on our previous effort of implementing a framework for the pi-calculus in

Isabelle using the nominal data type package, and strengthen our claim that this framework is well suited

to represent the theory of the pi-calculus, especially in the smooth treatment of bound names.