Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1/Reverse Implication
< Rule of Distribution | Disjunction Distributes over Conjunction | Left Distributive | Formulation 1
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Definition
- $\paren {p \lor q} \land \paren {p \lor r} \vdash p \lor \paren {q \land r}$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\paren {p \lor q} \land \paren {p \lor r}$ | Premise | (None) | ||
| 2 | 1 | $p \lor r$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
| 3 | 3 | $p$ | Assumption | (None) | ||
| 4 | 3 | $p \lor \paren {q \land r}$ | Rule of Addition: $\lor \II_1$ | 3 | ||
| 5 | 5 | $r$ | Assumption | (None) | ||
| 6 | 1 | $p \lor q$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
| 7 | 7 | $p$ | Assumption | (None) | ||
| 8 | 7 | $p \lor \paren {q \land r}$ | Rule of Addition: $\lor \II_1$ | 7 | ||
| 9 | 9 | $q$ | Assumption | (None) | ||
| 10 | 5, 9 | $q \land r$ | Rule of Conjunction: $\land \II$ | 9, 5 | ||
| 11 | 5, 9 | $p \lor \paren {q \land r}$ | Rule of Addition: $\lor \II_2$ | 10 | ||
| 12 | 1, 5 | $p \lor \paren {q \land r}$ | Proof by Cases: $\text{PBC}$ | 6, 7 – 8, 9 – 11 | Assumptions 7 and 9 have been discharged | |
| 13 | 1 | $p \lor \paren {q \land r}$ | Proof by Cases: $\text{PBC}$ | 2, 3 – 4, 5 – 12 | Assumptions 3 and 5 have been discharged |
$\blacksquare$