Factor Principles/Conjunction on Right/Formulation 1/Proof 2

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Theorem

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


Proof

By the tableau method of natural deduction:

$p \implies q \vdash \paren {p \land r} \implies \paren {q \land r} $
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Premise (None)
2 2 $p \land r$ Assumption (None)
3 2 $p$ Rule of Simplification: $\land \mathcal E_1$ 2
4 1, 2 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 3
5 2 $r$ Rule of Simplification: $\land \mathcal E_2$ 2
6 1, 2 $q \land r$ Rule of Conjunction: $\land \mathcal I$ 4, 5
7 1 $\paren {p \land r} \implies \paren {q \land r}$ Rule of Implication: $\implies \mathcal I$ 2 – 6 Assumption 2 has been discharged

$\blacksquare$


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