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

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$\paren {p \implies q} \land \paren {p \implies r} \vdash p \implies \paren {q \land r}$


By the tableau method of natural deduction:

$\paren {p \implies q} \land \paren {p \implies r} \vdash p \implies \paren {q \land r} $
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {p \implies q} \land \paren {p \implies r}$ Premise (None)
2 1 $\paren {p \land p} \implies \paren {q \land r}$ 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 $p \implies \paren {q \land r}$ Rule of Implication: $\implies \II$ 3 – 5 Assumption 3 has been discharged