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

Definition

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

Proof

 * align="right" | 2 ||
 * align="right" | 1
 * $p$
 * $\land \mathcal E_1$
 * 1
 * align="right" | 3 ||
 * align="right" | 1
 * $q \lor r$
 * $\land \mathcal E_2$
 * 1
 * $q \lor r$
 * $\land \mathcal E_2$
 * 1


 * align="right" | 6 ||
 * align="right" | 1, 4
 * $\left({p \land q}\right) \lor \left({p \land r}\right)$
 * $\lor \mathcal I_1$
 * 5
 * 5


 * align="right" | 9 ||
 * align="right" | 1, 7
 * $\left({p \land q}\right) \lor \left({p \land r}\right)$
 * $\lor \mathcal I_1$
 * 8
 * align="right" | 10 ||
 * align="right" | 1
 * $\left({p \land q}\right) \lor \left({p \land r}\right)$
 * $\lor \mathcal{E}$
 * 3, 4-6, 7-9
 * $\left({p \land q}\right) \lor \left({p \land r}\right)$
 * $\lor \mathcal{E}$
 * 3, 4-6, 7-9