# Rule of Exportation/Formulation 2

## Theorem

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

This can be expressed as two separate theorems:

### Forward Implication

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

### Reverse Implication

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

## Proof

### Proof of Forward Implication

By the tableau method of natural deduction:

$\vdash \paren {\paren {p \land q} \implies r} \implies \paren {p \implies \paren {q \implies r} }$
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {p \land q} \implies r$ Assumption (None)
2 1 $p \implies \paren {q \implies r}$ Sequent Introduction 1 Rule of Exportation: Forward Implication: Formulation 1
3 $\paren {\paren {p \land q} \implies r} \implies \paren {p \implies \paren {q \implies r} }$ Rule of Implication: $\implies \II$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$

### Proof of Reverse Implication

By the tableau method of natural deduction:

$\vdash \paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \land q} \implies r}$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies \paren {q \implies r}$ Assumption (None)
2 1 $\paren {p \land q} \implies r$ Sequent Introduction 1 Rule of Exportation: Reverse Implication: Formulation 1
3 $\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \land q} \implies r}$ Rule of Implication: $\implies \II$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$