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

Definition

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

Proof

 * align="right" | 3 ||
 * align="right" | 2
 * $p$
 * $\land \mathcal E_1$
 * 2
 * 2


 * align="right" | 5 ||
 * align="right" | 4
 * $p$
 * $\land \mathcal E_2$
 * 4
 * align="right" | 6 ||
 * align="right" | 1
 * $p$
 * $\lor \mathcal E$
 * 1, 2-3, 4-5
 * $p$
 * $\lor \mathcal E$
 * 1, 2-3, 4-5


 * align="right" | 8 ||
 * align="right" | 7
 * $q$
 * $\land \mathcal E_2$
 * 7
 * align="right" | 9 ||
 * align="right" | 7
 * $q \lor r$
 * $\lor \mathcal I_1$
 * 7
 * $q \lor r$
 * $\lor \mathcal I_1$
 * 7


 * align="right" | 11 ||
 * align="right" | 10
 * $r$
 * $\land \mathcal E_2$
 * 10
 * align="right" | 12 ||
 * align="right" | 10
 * $q \lor r$
 * $\lor \mathcal I_2$
 * 11
 * align="right" | 13 ||
 * align="right" | 1
 * $q \lor r$
 * $\lor \mathcal E$
 * 1, 7-9, 10-12
 * align="right" | 13 ||
 * align="right" | 1
 * $q \lor r$
 * $\lor \mathcal E$
 * 1, 7-9, 10-12