Rule of Transposition/Variant 1/Formulation 1/Forward Implication
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Theorem
- $p \implies \neg q \vdash q \implies \neg p$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \implies \neg q$ | Premise | (None) | ||
2 | 2 | $q$ | Assumption | (None) | ||
3 | 2 | $\neg \neg q$ | Double Negation Introduction: $\neg \neg \II$ | 2 | ||
4 | 1, 2 | $\neg p$ | Modus Tollendo Tollens (MTT) | 1, 3 | ||
5 | 1 | $q \implies \neg p$ | Rule of Implication: $\implies \II$ | 2 – 4 | Assumption 2 has been discharged |
$\blacksquare$