Definition 20: We recursively define when a statement is a substatement (or a component of) another statement.
(a) Base cases: Every statement is a substatement of itself.
(b) Recursion: If \xi is a substatement of at least one of \phi and \psi , then \xi is a substatement of the conjunction (\phi \wedge \psi).
(c) Recursion: If \xi is a substatement of \phi , then \xi is a substatement of the negation \neg \phi .
(d) Recursion: If \xi is a substatement of \phi , then \xi is a substatement of the existence statement \exists x \phi (where x is some variable).