Modus Tollendo Ponens/Sequent Form/Case 2

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Theorem

\(\ds p \lor q\) \(\) \(\ds \)
\(\ds \neg q\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds p\) \(\) \(\ds \)


Proof

By the tableau method of natural deduction:

$p \lor q, \neg q \vdash p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor q$ Premise (None)
2 2 $\neg q$ Premise (None)
3 3 $q$ Assumption (None)
4 2 $q \implies p$ Sequent Introduction 2 False Statement implies Every Statement
5 2, 3 $p$ Modus Ponendo Ponens: $\implies \mathcal E$ 4, 3
6 6 $p$ Assumption (None)
7 1, 2 $p$ Proof by Cases: $\text{PBC}$ 1, 3 – 5, 6 – 6 Assumptions 3 and 6 have been discharged

$\blacksquare$


Sources