# Hypothetical Syllogism/Formulation 2/Proof 2

## Theorem

 $\ds p$ $\implies$ $\ds q$ $\ds q$ $\implies$ $\ds r$ $\ds r$  $\ds$ $\ds \vdash \ \$ $\ds p$  $\ds$

## Proof

By the tableau method of natural deduction:

$p \implies q, q \implies r, p \vdash r$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Premise (None)
2 2 $q \implies r$ Premise (None)
3 3 $p$ Premise (None)
4 1, 3 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 3
5 1, 2, 3 $r$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 4

$\blacksquare$