Hypothetical Syllogism/Formulation 1/Proof 1

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Theorem

$p \implies q, q \implies r \vdash p \implies r$


Proof

By the tableau method of natural deduction:

$p \implies q, q \implies r \vdash p \implies r$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Premise (None)
2 2 $q \implies r$ Premise (None)
3 3 $p$ Assumption (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
6 1, 2 $p \implies r$ Rule of Implication: $\implies \mathcal I$ 3 – 5 Assumption 3 has been discharged

$\blacksquare$