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

Definition

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

$p \land \paren {q \lor r} \vdash \paren {p \land q} \lor \paren {p \land r} $
Line	Pool	Formula	Rule	Depends upon	Notes
1	1	$p \land \paren {q \lor r}$	Premise	(None)
2	1	$p$	Rule of Simplification: $\land \EE_1$	1
3	1	$q \lor r$	Rule of Simplification: $\land \EE_2$	1
4	4	$q$	Assumption	(None)
5	1, 4	$p \land q$	Rule of Conjunction: $\land \II$	2, 4
6	1, 4	$\paren {p \land q} \lor \paren {p \land r}$	Rule of Addition: $\lor \II_1$	5
7	7	$r$	Assumption	(None)
8	1, 7	$p \land r$	Rule of Conjunction: $\land \II$	2, 7
9	1, 7	$\paren {p \land q} \lor \paren {p \land r}$	Rule of Addition: $\lor \II_2$	8
10	1	$\paren {p \land q} \lor \paren {p \land r}$	Proof by Cases: $\text{PBC}$	3, 4 – 6, 7 – 9	Assumptions 4 and 7 have been discharged

$\blacksquare$