Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2/Reverse Implication

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Theorem

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


Proof

By the tableau method of natural deduction:

$\vdash \paren {\paren {p \land q} \lor \paren {p \land r} } \implies \paren {p \land \paren {q \lor r} } $
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {p \land q} \lor \paren {p \land r}$ Assumption (None)
2 1 $p \land \paren {q \lor r}$ Sequent Introduction 1 Conjunction is Left Distributive over Disjunction: Formulation 1
3 $\paren {\paren {p \land q} \lor \paren {p \land r} } \implies \paren {p \land \paren {q \lor r} }$ Rule of Implication: $\implies \II$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$