For every ccc partial ordering \(P\), if an existential statement about \(HC\) holds in an (ideal) generic extension of \(V\) obtained by forcing with \(P\), then the statement is true, i.e., it holds in \(V\). In other words, if a set having a property that depends only on sets in \(HC\) exists in some (ideal) generic extension of \(V\) obtained by forcing with a ccc partial ordering, then a set with that property already exists in \(V\).