Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 1/Forward Implication
< Rule of Distribution | Disjunction Distributes over Conjunction | Right Distributive | Formulation 1
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Definition
- $\paren {q \land r} \lor p \vdash \paren {q \lor p} \land \paren {r \lor p}$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\paren {q \land r} \lor p$ | Premise | (None) | ||
| 2 | 1 | $p \lor \paren {q \land r}$ | Sequent Introduction | 1 | Disjunction is Commutative | |
| 3 | 1 | $\paren {p \lor q} \land \paren {p \lor r}$ | Sequent Introduction | 2 | Disjunction is Left Distributive over Conjunction | |
| 4 | 1 | $\paren {q \lor p} \land \paren {r \lor p}$ | Sequent Introduction | 3 | Disjunction is Commutative |
$\blacksquare$