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

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## Theorem

- $p \implies \paren {q \land r} \vdash \paren {p \implies q} \land \paren {p \implies r}$

## Proof

By the tableau method of natural deduction:

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

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

2 | 2 | $p$ | Assumption | (None) | ||

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

4 | 1, 2 | $q$ | Rule of Simplification: $\land \EE_1$ | 3 | ||

5 | 1, 2 | $r$ | Rule of Simplification: $\land \EE_2$ | 3 | ||

6 | 1 | $p \implies q$ | Rule of Implication: $\implies \II$ | 2 – 4 | Assumption 2 has been discharged | |

7 | 1 | $p \implies r$ | Rule of Implication: $\implies \II$ | 2 – 5 | Assumption 2 has been discharged | |

8 | 1 | $\paren {p \implies q} \land \paren {p \implies r}$ | Rule of Conjunction: $\land \II$ | 6, 7 |

$\blacksquare$