Implication is Left Distributive over Conjunction/Reverse Implication/Formulation 1/Proof

Theorem

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

Proof

 * align="right" | 2 ||
 * align="right" | 1
 * $\left({p \land p}\right) \implies \left({q \land r}\right)$
 * Sequent Introduction
 * 1
 * Praeclarum Theorema
 * Praeclarum Theorema


 * align="right" | 4 ||
 * align="right" | 3
 * $p \land p$
 * Sequent Introduction
 * 3
 * Rule of Idempotence
 * align="right" | 5 ||
 * align="right" | 1, 3
 * $q \land r$
 * Modus Ponendo Ponens
 * 2, 4
 * align="right" | 6 ||
 * align="right" | 1
 * $p \implies \left({q \land r}\right)$
 * Rule of Implication
 * 3, 5
 * align="right" | 1
 * $p \implies \left({q \land r}\right)$
 * Rule of Implication
 * 3, 5