# Rule of Material Implication/Formulation 2/Reverse Implication

## Theorem

$\vdash \paren {\neg p \lor q} \implies \paren {p \implies q}$

## Proof

By the tableau method of natural deduction:

$\paren {\neg p \lor q} \implies \paren {p \implies q}$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p \lor q$ Assumption (None)
2 1 $p \implies q$ Sequent Introduction 1 Rule of Material Implication: Formulation 1
3 $\paren {\neg p \lor q} \implies \paren {p \implies q}$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$