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

Theorem

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

Proof

 * align="right" | 3 ||
 * align="right" | 1, 2
 * $q \land r$
 * Modus Ponendo Ponens
 * 1, 2
 * align="right" | 4 ||
 * align="right" | 1, 2
 * $q$
 * Rule of Simplification
 * 3
 * align="right" | 5 ||
 * align="right" | 1, 2
 * $r$
 * Rule of Simplification
 * 3
 * align="right" | 6 ||
 * align="right" | 1
 * $p \implies q$
 * Rule of Implication
 * 2, 4
 * align="right" | 7 ||
 * align="right" | 1
 * $p \implies r$
 * Rule of Implication
 * 2, 5
 * align="right" | 8 ||
 * align="right" | 1
 * $\left({p \implies q}\right) \land \left({p \implies r}\right)$
 * Rule of Conjunction
 * 6, 7
 * }
 * $p \implies r$
 * Rule of Implication
 * 2, 5
 * align="right" | 8 ||
 * align="right" | 1
 * $\left({p \implies q}\right) \land \left({p \implies r}\right)$
 * Rule of Conjunction
 * 6, 7
 * }
 * 6, 7
 * }
 * }