Rule of Distribution/Conjunction Distributes over Disjunction/Right Distributive/Formulation 1/Forward Implication

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Definition

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


Proof

By the tableau method of natural deduction:

$\paren {q \lor r} \land p \vdash \paren {q \land p} \lor \paren {r \land p} $
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {q \lor r} \land p$ Premise (None)
2 1 $p \land \paren {q \lor r}$ Sequent Introduction 1 Conjunction is Commutative
3 1 $\paren {p \lor q} \land \paren {p \lor r}$ Sequent Introduction 2 Conjunction is Left Distributive over Disjunction
4 1 $\paren {q \land p} \lor \paren {r \land p}$ Sequent Introduction 3 Conjunction is Commutative

$\blacksquare$