Conjunction implies Disjunction of Conjunctions with Complements

Theorem

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

Proof

 * align="right" | 2 ||
 * align="right" | -
 * $r \lor \neg r$
 * LEM
 * (None)
 * align="right" | 3 ||
 * align="right" | 1
 * $p$
 * $\land \mathcal E_1$
 * 1
 * align="right" | 4 ||
 * align="right" | 1
 * $q$
 * $\land \mathcal E_2$
 * 1
 * align="right" | 4 ||
 * align="right" | 1
 * $q$
 * $\land \mathcal E_2$
 * 1


 * align="right" | 6 ||
 * align="right" | 1, 5
 * $p \land r$
 * $\land \mathcal I$
 * 1, 5
 * align="right" | 7 ||
 * align="right" | 1, 5
 * $\left({p \land r}\right) \lor \left({q \land \neg r}\right)$
 * $\lor \mathcal I_1$
 * 6
 * $\left({p \land r}\right) \lor \left({q \land \neg r}\right)$
 * $\lor \mathcal I_1$
 * 6


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