Rule of Implication/Proof Rule

Proof Rule
The rule of implication is a valid argument in types of logic dealing with conditionals $\implies$.

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

As a proof rule it is expressed in the form:
 * If, by making an assumption $\phi$, we can conclude $\psi$ as a consequence, we may infer $\phi \implies \psi$.


 * The conclusion $\phi \implies \psi$ does not depend on the assumption $\phi$, which is thus discharged.

It can be written:
 * $\ds {\begin {array} {|c|} \hline \phi \\ \vdots \\ \psi \\ \hline \end {array} \over \phi \implies \psi} \to_i$