Rule of Implication/Sequent Form/Proof 1

Theorem

The Rule of Implication can be symbolised by the sequent:

\(\ds \paren {p \vdash q}\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds p \implies q\) \(\) \(\ds \)


Proof

By the tableau method of natural deduction:

$\paren {p \vdash q} \vdash p \implies q$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Premise (None)
2 1 $q$ By hypothesis 1 as $p \vdash q$
3 1 $p \implies q$ Rule of Implication: $\implies \II$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$

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