# Implication is Left Distributive over Conjunction/Reverse Implication/Formulation 1/Proof

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## Theorem

- $\left({p \implies q}\right) \land \left({p \implies r}\right) \vdash p \implies \left({q \land r}\right)$

## Proof

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\left({p \implies q}\right) \land \left({p \implies r}\right)$ | Premise | (None) | ||

2 | 1 | $\left({p \land p}\right) \implies \left({q \land r}\right)$ | Sequent Introduction | 1 | Praeclarum Theorema | |

3 | 3 | $p$ | Assumption | (None) | ||

4 | 3 | $p \land p$ | Sequent Introduction | 3 | Rule of Idempotence: Conjunction | |

5 | 1, 3 | $q \land r$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 2, 4 | ||

6 | 1 | $p \implies \left({q \land r}\right)$ | Rule of Implication: $\implies \mathcal I$ | 3 – 5 | Assumption 3 has been discharged |

$\blacksquare$