![](Pasted%20image%2020250221132524.png)

| Line |                             |            |                                   |
| ---- | --------------------------- | ---------- | --------------------------------- |
| 1    | $A \implies \lnot B$        | Premise    |                                   |
| 2    | $(B \lor C) \lor D$         | Premise    |                                   |
| 3    | $\lnot C \lor D \implies A$ | Premise    |                                   |
| 4    | $\lnot C$                   | Premise    |                                   |
| 5    | $\lnot C \lor D$            | From 4     | $\lor Intro$                      |
| 6    | $A$                         | From 3 & 5 | $\implies Elim$ Modus Ponens      |
| 7    | $\lnot B$                   | From 1 & 6 | $\implies Elim$ Modus Ponens      |
| 8    | $\lnot B \land \lnot C$     | From 2 & 7 | $\land Intro$                     |
| 9    | $\lnot (B \lor C)$          | From 8     | De Morgans Law                    |
| 10   | $D$                         | From 9     | $\lor Elim$ Disjunctive Syllogism |
B or C is not true, therefore D must be true from step 2.