Implication is Left Distributive over Conjunction/Reverse Implication/Formulation 1/Proof

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Theorem

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


Proof

By the tableau method of natural deduction:

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

$\blacksquare$