Constructive Dilemma/Formulation 1/Proof 1

Theorem

 * $p \implies q, r \implies s \vdash p \lor r \implies q \lor s$

Proof

 * align="right" | 5 ||
 * align="right" | 1, 4
 * $q$
 * $\implies \mathcal E$
 * 1, 4
 * align="right" | 6 ||
 * align="right" | 1, 4
 * $q \lor s$
 * $\lor \mathcal I_1$
 * 5
 * $q \lor s$
 * $\lor \mathcal I_1$
 * 5


 * align="right" | 8 ||
 * align="right" | 2, 7
 * $s$
 * $\implies \mathcal E$
 * 2, 7
 * align="right" | 9 ||
 * align="right" | 2, 7
 * $q \lor s$
 * $\lor \mathcal I_2$
 * 8
 * align="right" | 10 ||
 * align="right" | 1, 2, 3
 * $q \lor s$
 * $\lor \mathcal E$
 * 3, 4-6, 7-9
 * align="right" | 11 ||
 * align="right" | 1, 2
 * $p \lor r \implies q \lor s$
 * $\implies \mathcal I$
 * 3, 10
 * }
 * align="right" | 11 ||
 * align="right" | 1, 2
 * $p \lor r \implies q \lor s$
 * $\implies \mathcal I$
 * 3, 10
 * }
 * }
 * }