Implication is Left Distributive over Disjunction/Formulation 1/Reverse Implication
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Theorem
- $\paren {p \implies q} \lor \paren{p \implies r} \vdash p \implies \paren {q \lor r}$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\paren {p \implies q} \lor \paren {p \implies r}$ | Assumption | (None) | ||
| 2 | 2 | $p \implies q$ | Assumption | (None) | ||
| 3 | 3 | $p$ | Assumption | (None) | ||
| 4 | 2, 3 | $q$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 2, 3 | ||
| 5 | 2, 3 | $q \lor r$ | Rule of Addition: $\lor \II_1$ | 4 | ||
| 6 | 2 | $p \implies \paren {q \lor r}$ | Rule of Implication: $\implies \II$ | 3 – 5 | Assumption 3 has been discharged | |
| 7 | 7 | $p \implies r$ | Assumption | (None) | ||
| 8 | 8 | $p$ | Assumption | (None) | ||
| 9 | 7, 8 | $r$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 7, 8 | ||
| 10 | 7, 8 | $q \lor r$ | Rule of Addition: $\lor \II_2$ | 9 | ||
| 11 | 7 | $p \implies \paren {q \lor r}$ | Rule of Implication: $\implies \II$ | 8 – 10 | Assumption 8 has been discharged | |
| 12 | 1 | $p \implies \paren{q \lor r}$ | Proof by Cases: $\text{PBC}$ | 1, 2 – 6, 7 – 11 | Assumptions 2 and 7 have been discharged |
$\blacksquare$