Rule of Transposition/Formulation 2/Forward Implication/Proof

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Theorem



$\vdash \paren {p \implies q} \implies \paren {\neg q \implies \neg p}$


Proof

By the tableau method of natural deduction:

$\vdash \paren {p \implies q} \implies \paren {\neg q \implies \neg p} $
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Assumption (None)
2 2 $\neg q$ Assumption (None)
3 1, 2 $\neg p$ Modus Tollendo Tollens (MTT) 1, 2
4 1 $\neg q \implies \neg p$ Rule of Implication: $\implies \II$ 2 – 3 Assumption 2 has been discharged
5 $\paren {p \implies q} \implies \paren {\neg q \implies \neg p}$ Rule of Implication: $\implies \II$ 1 – 4 Assumption 1 has been discharged

$\blacksquare$


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