  We denote by Conc(L) the semilattice of all finitely generated congruences of
a lattice L. For varieties (i.e., equational classes) V and W of lattices such
that V is contained neither in W nor its dual, and such that every simple
member of W contains a prime interval, we prove that there exists a bounded
lattice A in V with at most aleph 2 elements such that Conc(A) is not
isomorphic to Conc(B) for any B in W. The bound aleph 2 is optimal. As a
corollary of our results, there are continuum many congruence classes of
locally finite varieties of (bounded) modular lattices.
