 Solution

Here is the existential common-substructure test for quantifier elimination. Use a first-order language with a constant symbol, as in the applications below, or the usual conventions for empty generated substructures and truth constants. Let $T$ be a first-order theory. Suppose that whenever $\mathcal{M},\mathcal{N}\models T$ have a common substructure $\mathcal{A}$, then, for every quantifier-free formula $\theta (x,\bar{y})$ and every finite tuple $\bar{a}\in A$,The condition is symmetric because it applies also with $\mathcal{M}$ and $\mathcal{N}$ exchanged. The conclusion is quantifier elimination for $T$: for every first-order formula $\phi (\bar{y})$ there is a quantifier-free formula $\psi (\bar{y})$ such thatThe common substructure need not itself satisfy $T$, and witnesses are sought in the whole models, not necessarily in $A$. This test is the QET1 version of the common-substructure test for quantifier elimination.