Rule of Exportation/Forward Implication/Formulation 1/Proof

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Theorem

$\left ({p \land q}\right) \implies r \vdash p \implies \left ({q \implies r}\right)$


Proof

By the tableau method of natural deduction:

$\left ({p \land q}\right) \implies r \vdash p \implies \left ({q \implies r}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\left ({p \land q}\right) \implies r$ Premise (None)
2 2 $p$ Assumption (None)
3 3 $q$ Assumption (None)
4 2, 3 $p \land q$ Rule of Conjunction: $\land \mathcal I$ 2, 3
5 1, 2, 3 $r$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 4
6 1, 2 $q \implies r$ Rule of Implication: $\implies \mathcal I$ 3 – 5 Assumption 3 has been discharged
7 1 $p \implies \left ({q \implies r}\right)$ Rule of Implication: $\implies \mathcal I$ 2 – 6 Assumption 2 has been discharged

$\blacksquare$


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