Database: Variable occurrence in equivalences

Proposition 17: If a variable x occurs in at least one of the statements \phi and \psi , then it occurs in the equivalence (\phi \Leftrightarrow \psi) .

Proof: By [16], x occurs in the implications (\phi \Rightarrow \psi) and (\psi \Rightarrow \phi) . Thus, by (d) in the definition of occurrence, x occurs in the conjunction \big( (\phi \Rightarrow \psi) \wedge (\psi \Rightarrow \phi) \big). But (\phi \Leftrightarrow \psi) is short for precisely this statement. \square

This database entry builds on the following:

  1. Terminology: Class, membership
  2. Terminology: Statement
  3. Definition: Conjunction
  4. Definition: Occurrence of a variable
  5. Definition: Implication
  6. Proposition: Variable occurrence in implications
  7. Definition: Equivalence