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

## Definition

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

## Proof

By the tableau method of natural deduction:

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

$\blacksquare$