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

Definition

 * $p \lor \left({q \land r}\right) \vdash \left({p \lor q}\right) \land \left({p \lor r}\right)$

Proof

 * align="right" | 3 ||
 * align="right" | 2
 * $p \lor q$
 * $\lor \mathcal I_1$
 * 2
 * align="right" | 4 ||
 * align="right" | 2
 * $p \lor r$
 * $\lor \mathcal I_1$
 * 2
 * $p \lor r$
 * $\lor \mathcal I_1$
 * 2


 * align="right" | 7 ||
 * align="right" | 6
 * $q$
 * $\land \mathcal E_1$
 * 6
 * align="right" | 8 ||
 * align="right" | 6
 * $r$
 * $\land \mathcal E_2$
 * 6
 * align="right" | 9 ||
 * align="right" | 6
 * $p \lor q$
 * $\lor \mathcal I_2$
 * 7
 * align="right" | 10 ||
 * align="right" | 6
 * $p \lor r$
 * $\lor \mathcal I_2$
 * 8
 * align="right" | 10 ||
 * align="right" | 6
 * $p \lor r$
 * $\lor \mathcal I_2$
 * 8
 * $\lor \mathcal I_2$
 * 8


 * align="right" | 12 ||
 * align="right" | 1
 * $\left({p \land q}\right) \lor \left({p \land r}\right)$
 * $\lor \mathcal E$
 * 1, 2-5, 6-11
 * 1, 2-5, 6-11