This paper provides a starting point for generic quantifier elimination by partial cylindrical algebraic decomposition (\textsc{pcad}). On input of a first-order formula over the reals generic \textsc{pcad} outputs a theory and a quantifier-free formula. The theory is a set of negated equations in the free variables of the input formula. The quantifier-free formula is equivalent to the input for all parameter values satisfying the theory. For obtaining this generic elimination procedure, we derive a generic projection operator from the standard Collins--Hong projection operator. Our operator particularly addresses formulas with many parameters thus filling a gap in the applicability of \textsc{pcad}. It restricts decomposition to a reasonable subset of the entire space. The above-mentioned theory describes this subset. The approach is compatible with other improvements in the framework of \textsc{pcad}. It turns out that the theory contains assumptions that are easily interpretable and that are most often non-degeneracy conditions. The applicability of our generic elimination procedure significantly extends that of the corresponding regular procedure. Our procedure is implemented in the computer logic system \textsc{redlog}.