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

Definition

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

$\paren {p \land q} \lor \paren {p \land r} \vdash p \land \paren {q \lor r} $
Line	Pool	Formula	Rule	Depends upon	Notes
1	1	$\paren {p \land q} \lor \paren {p \land r}$	Premise	(None)
2	2	$p \land q$	Assumption	(None)
3	2	$p$	Rule of Simplification: $\land \EE_1$	2
4	4	$p \land r$	Assumption	(None)
5	4	$p$	Rule of Simplification: $\land \EE_1$	2
6	1	$p$	Proof by Cases: $\text{PBC}$	1, 2 – 3, 4 – 5	Assumptions 2 and 4 have been discharged
7	7	$p \land q$	Assumption	(None)
8	7	$q$	Rule of Simplification: $\land \EE_2$	7
9	7	$q \lor r$	Rule of Addition: $\lor \II_1$	7
10	10	$p \land r$	Assumption	(None)
11	10	$r$	Rule of Simplification: $\land \EE_2$	10
12	10	$q \lor r$	Rule of Addition: $\lor \II_2$	11
13	1	$q \lor r$	Proof by Cases: $\text{PBC}$	1, 7 – 9, 10 – 12	Assumptions 7 and 10 have been discharged
14	1	$p \land \paren {q \lor r}$	Rule of Conjunction: $\land \II$	6, 13

$\blacksquare$