Modus Ponendo Ponens/Proof Rule

Proof Rule
Modus ponendo ponens 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 we can conclude $\phi \implies \psi$, and we can also conclude $\phi$, then we may infer $\psi$.

Thus it provides a means of eliminating a conditional from a sequent.

It can be written:
 * $\ds {\phi \qquad \phi \implies \psi \over \psi} \to_e$

Also see

 * This is a rule of inference of the following proof systems:
 * Definition:Natural Deduction
 * Definition:Hilbert Proof System/Predicate Logic/Instance 1
 * Definition:Hilbert Proof System/Instance 1
 * Definition:Hilbert Proof System/Instance 2