# Rule of Exportation/Reverse Implication/Formulation 1/Proof

## Theorem

$p \implies \paren {q \implies r} \vdash \paren {p \land q} \implies r$

## Proof

By the tableau method of natural deduction:

$p \implies \paren {q \implies r} \vdash \paren {p \land q} \implies r$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies \paren {q \implies r}$ Premise (None)
2 2 $p \land q$ Assumption (None)
3 2 $p$ Rule of Simplification: $\land \EE_1$ 2
4 2 $q$ Rule of Simplification: $\land \EE_2$ 2
5 1, 2 $q \implies r$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 3
6 1, 2 $r$ Modus Ponendo Ponens: $\implies \mathcal E$ 5, 4
7 1 $\paren {p \land q} \implies r$ Rule of Implication: $\implies \II$ 2 – 6 Assumption 2 has been discharged

$\blacksquare$