Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 2
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Theorem
The disjunction operator is right distributive over the conjunction operator:
- $\vdash \paren {\paren {q \land r} \lor p} \iff \paren {\paren {q \lor p} \land \paren {r \lor p} }$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $\paren {q \land r} \lor p$ | Assumption | (None) | ||
2 | 1 | $\paren {q \lor p} \land \paren {r \lor p}$ | Sequent Introduction | 1 | Conjunction is Right Distributive over Disjunction: Formulation 1 | |
3 | $\paren {\paren {q \land r} \lor p} \implies \paren {\paren {q \lor p} \land \paren {r \lor p} }$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
4 | 4 | $\paren {q \lor p} \land \paren {r \lor p}$ | Assumption | (None) | ||
5 | 4 | $\paren {q \land r} \lor p$ | Sequent Introduction | 4 | Conjunction is Right Distributive over Disjunction: Formulation 1 | |
6 | $\paren {\paren {q \lor p} \land \paren {r \lor p} } \implies \paren {\paren {q \land r} \lor p}$ | Rule of Implication: $\implies \II$ | 4 – 5 | Assumption 4 has been discharged | ||
7 | $\paren {\paren {q \land r} \lor p} \iff \paren {\paren {q \lor p} \land \paren {r \lor p} }$ | Biconditional Introduction: $\iff \II$ | 3, 6 |
$\blacksquare$