  Mellin-Barnes (MB) representations have become a widely used tool for the
evaluation of Feynman loop integrals appearing in perturbative calculations of
quantum field theory. Some of the MB integrals may be solved analytically in
closed form with the help of the two Barnes lemmas which have been known in
mathematics already for one century. The original proofs of these lemmas solve
the integrals by taking infinite series of residues and summing these up via
hypergeometric functions. This paper presents new, elegant proofs for the
Barnes lemmas which only rely on the well-known basic identity of MB
representations, avoiding any series summations. They are particularly useful
for presenting and proving the Barnes lemmas to students of quantum field
theory without requiring knowledge on hypergeometric functions. The paper also
introduces and proves an additional lemma for a MB integral \int dz involving a
phase factor exp(+-i pi z).
