# Contradictory Antecedent/Proof 1

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## Theorem

- $\bot \implies p \dashv \vdash \top$

## Proof

By the tableau method of natural deduction:

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

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

2 | $\top$ | Rule of Top-Introduction: $\top \II$ | (None) |

$\Box$

By the tableau method of natural deduction:

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

1 | 1 | $\bot$ | Assumption | (None) | ||

2 | 2 | $\top$ | Premise | (None) | ||

3 | 1 | $p$ | Rule of Explosion: $\bot \EE$ | 1 | ||

4 | $\bot \implies p$ | Rule of Implication: $\implies \II$ | 1 – 3 | Assumption 1 has been discharged |

$\blacksquare$