# Rule of Association/Disjunction/Formulation 2/Proof 2

## Theorem

$\vdash \paren {p \lor \paren {q \lor r} } \iff \paren {\paren {p \lor q} \lor r}$

## Proof

This proof is derived in the context of the following proof system: Instance 2 of the Hilbert-style systems.

By the tableau method:

$\vdash \paren {p \lor \paren {q \lor r} } \iff \paren {\paren {p \lor q} \lor r}$
Line Pool Formula Rule Depends upon Notes
1 $\paren {p \lor \paren {q \lor r} } \implies \paren {\paren {p \lor q} \lor r}$ Rule of Association: Forward Implication
2 $\paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} }$ Rule of Association: Reverse Implication
3 $\paren {\paren {p \lor \paren {q \lor r} } \implies \paren {\paren {p \lor q} \lor r} } \land \paren {\paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} } }$ Rule $\text {RST} 4$ 1, 2
4 $\paren {p \lor \paren {q \lor r} } \iff \paren {\paren {p \lor q} \lor r}$ Rule $\text {RST} 2 (3)$ 3

$\blacksquare$