Demonstrate the next propositional excersise( The Constructive Dilema)

Question

How do I solve the next exercise using inference laws, This Rule is also called, "Constructive Dilema"

\begin{align} p & \rightarrow q \\ -p & \rightarrow -q \\ p & \lor -p \\ \hline \\ q & \lor -q \end{align}

Answer 1

Here is a proof that just uses the first two premises:

1. $p \rightarrow q$ Premise

2. $\neg p \rightarrow \neg q$ Premise

3. $\neg p \lor q$ Conditional 1

4. $q \lor \neg p$ Commutative 3

5. $\neg \neg q \lor \neg p$ Double Negative 4

6. $\neg q \rightarrow \neg p$ Conditional 5

7. $\neg q \rightarrow \neg q$ Transitivity 2,6

8. $\neg \neg q \lor \neg q$ Conditional 7

9. $q \lor \neg q$ Double Negative 8

And here is a (much simpler!) proof that uses just the third premise:

1. $p \lor \neg p$ Premise

2. $\top$ Negation 1

3. $q \lor \neg q$ Negation 2