 In order to state the definition of equinumerosity, we shall employ the well-known logical notion ‘there exists a unique \(x\) such that \(\phi\)’. To say that there exists a unique \(x\) such that \(\phi\) is to say: there is some \(x\) such that \(\phi\), and anything \(y\) which is such that \(\phi\) is identical to \(x\). In what follows, we use the notation ‘\(\exists!x\phi\)’ to abbreviate this notion of a formula being uniquely satisfied, and we define it formally as follows (where again, \(\phi^y_x\) is the result of substituting \(y\) for the free occurrences of \(x\) in \(\phi\): 