Proof by Cases/Proof Rule

Proof Rule
Proof by cases is a valid argument in types of logic dealing with disjunctions $\lor$.

This includes propositional logic and predicate logic, and in particular natural deduction.

As a proof rule it is expressed in the form:
 * If we can conclude $\phi \lor \psi$, and:
 * $(1): \quad$ By making the assumption $\phi$, we can conclude $\chi$
 * $(2): \quad$ By making the assumption $\psi$, we can conclude $\chi$
 * then we may infer $\chi$.

The conclusion $\chi$ does not depend upon either assumption $\phi$ or $\psi$.

It can be written:
 * $\ds {\phi \lor \psi \quad \begin{array}{|c|} \hline \phi \\ \vdots \\ \chi \\ \hline \end{array} \quad \begin{array}{|c|} \hline \psi \\ \vdots \\ \chi \\ \hline \end{array} \over \chi} \lor_e$