  We investigate the elimination of quantifiers in first-order formulas via
Hilbert's epsilon-operator (or -binder), following Bernays' explicit
definitions of the existential and the universal quantifier symbol by means of
epsilon-terms. This elimination has its first explicit occurrence in the proof
of the first epsilon-theorem in Hilbert-Bernays in 1939. We think that there is
a lacuna in this proof w.r.t. this elimination, related to the erroneous
assumption that explicit definitions always terminate. Surprisingly, to the
best of our knowledge, nobody ever proved confluence or termination for this
elimination procedure. Even myths on non-confluence and the openness of the
termination problem are circulating. We show confluence and termination of this
elimination procedure by means of a direct, straightforward, and easily
verifiable proof, based on a new theorem on how to obtain termination from weak
normalization.
