# Rule of Implication

(Redirected from Rule of Conditional Proof)

## Proof Rule

The rule of implication is a valid deduction sequent in propositional logic.

### Proof Rule

If, by making an assumption $\phi$, we can conclude $\psi$ as a consequence, we may infer $\phi \implies \psi$.

### Sequent Form

The Rule of Implication can be symbolised by the sequent:

$\left({p \vdash q}\right) \vdash p \implies q$

## Explanation

The Rule of Implication can be expressed in natural language as:

If by making an assumption $\phi$ we can deduce $\psi$, then we can encapsulate this deduction into the compound statement $\phi \implies \psi$.

## Also known as

The Rule of Implication is sometimes known as:

• The rule of implies-introduction
• The rule of conditional proof (abbreviated $\text{CP}$).