# Hypothetical Syllogism/Formulation 2/Proof 1

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## Theorem

- $p \implies q, q \implies r, p \vdash r$

## Proof

By the tableau method of natural deduction:

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

1 | 1 | $p \implies q$ | Premise | (None) | ||

2 | 2 | $q \implies r$ | Premise | (None) | ||

3 | 3 | $p$ | Premise | (None) | ||

4 | 1, 2 | $p \implies r$ | Sequent Introduction | 1, 2 | Hypothetical Syllogism: Formulation 1 | |

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

$\blacksquare$