Rule of Transposition/Formulation 2/Forward Implication/Proof

From ProofWiki
Jump to navigation Jump to search


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


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