  We find Bethe vectors for quantum integrable models associated with the
supersymmetric Yangians $Y(\mathfrak{gl}(m|n)$ in terms of the current
generators of the Yangian double $DY(\mathfrak{gl}(m|n))$. More specifically,
we use the method of projections onto intersections of different type Borel
subalgebras in this infinite dimensional algebra to construct the Bethe
vectors. Calculating these projection the supersymmetric Bethe vectors can be
expressed through matrix elements of the universal monodromy matrix elements.
Using two different but isomorphic current realizations of the Yangian double
$DY(\mathfrak{gl}(m|n))$ we obtain two different presentations for the Bethe
vectors. These Bethe vectors are also shown to obey some recursion relations
which prove their equivalence.
