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

Definition

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

Proof

By the tableau method of natural deduction:

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

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

$\blacksquare$