# Rule of Exportation/Formulation 1/Proof 1

## Theorem

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

## Proof

### Proof of Forward Implication

By the tableau method of natural deduction:

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

$\blacksquare$

### Proof of Reverse Implication

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

$\blacksquare$