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

Definition

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

Proof

 * align="right" | 2 ||
 * align="right" | 1
 * $p \lor r$
 * $\land \mathcal E_2$
 * 1
 * 1


 * align="right" | 4 ||
 * align="right" | 3
 * $p \lor \left({q \land r}\right)$
 * $\lor \mathcal I_1$
 * 3
 * 3


 * align="right" | 6 ||
 * align="right" | 1
 * $p \lor q$
 * $\land \mathcal E_1$
 * 1
 * 1


 * align="right" | 8 ||
 * align="right" | 7
 * $p \lor \left({q \land r}\right)$
 * $\lor \mathcal I_1$
 * 7
 * 7


 * align="right" | 11 ||
 * align="right" | 5, 9
 * $p \lor \left({q \land r}\right)$
 * $\lor \mathcal I_2$
 * 10
 * align="right" | 12 ||
 * align="right" | 1, 5
 * $p \lor \left({q \land r}\right)$
 * $\lor \mathcal E$
 * 6, 7-8, 9-11
 * align="right" | 13 ||
 * align="right" | 1
 * $p \lor \left({q \land r}\right)$
 * $\lor \mathcal E$
 * 2, 3-4, 5-12
 * align="right" | 13 ||
 * align="right" | 1
 * $p \lor \left({q \land r}\right)$
 * $\lor \mathcal E$
 * 2, 3-4, 5-12