 The definition for \(\lnot\), for instance, may be justified by the following equivalences. \(A\Leftrightarrow A\land A\), \(A\land A\Leftrightarrow \exists x(A\land A)\) (assuming that \(x\) is not free in \(A\)), hence by replacement, \(\lnot A\Leftrightarrow \lnot\exists x(A\land A)\).