Conducting a Proof Sequence - Discrete Mathematics I am having trouble conducting this proof sequence. Here are the premises:
p -> (q v r)
~q
~r

~p ^ ~ r
So far, I have this:

*

*p -> (q v r) - Premise

*~q - Premise

*~r - Premise

*~(q v r) - implication, 1

*~q ^ ~r - De Morgan's Law, 4

*~r ^ ~q - commutativity, 5

*~r - simplification, 6

*p -> (~r) - ???

*~p - Modus Tollens, 8, 3

*~p ^ ~r - conjunction, 9, 3

Is this sequence correct? I have a hard time understanding the use of implication and when/when not to apply it. My thoughts are using it whenever the disjunction is seen. Any help is greatly appreciated. Thank you!
 A: You can't infer $\ {\sim}(q\,\vee r)\ $  from the first premise alone, as you try to do in step $4$ of your argument.  You can, however, infer $\ {\sim}q\,\wedge{\sim}r\ $ from the second and third premises by using conjunction introduction, and you can then infer $\ {\sim}(q\,\vee r)\ $ from De Morgan's laws.
Your step $9$ is also incorrect.  Modus tollens doesn't allow you to infer $\ {\sim}p\ $ from $\ p\rightarrow{\sim}r\ $ and $\ {\sim}r\ $.  You can, however use modus tollens to infer $\ {\sim}p\ $ from premise $1$ and $\ {\sim}(q\,\vee r)\ $, and this latter proposition can be validly inferred from the premises in the way explained above.
Steps $7$ and $8$ of your argument are both valid but redundant, both because $\ {\sim}r\ $ is already a premise, and because these two inferences are only used for the invalid step $9$.
Step $10$ of your argument becomes a valid inference once you've replaced your invalid deduction of $\ {\sim}p\ $ with a valid one.  So, if you adopt the corrections suggested above, you'll get the following valid argument:

*

*$\ p\rightarrow(q\vee r)\ $—premise.

*$\ {\sim}q\hspace{4em}$—premise.

*$\ {\sim}r\hspace{4em}$—premise.

*$\ {\sim}q\,\wedge{\sim}r\hspace{1.5em}$—conjunction introduction, $2,3$.

*$\ {\sim}(q\,\vee r)\hspace{1.5em}$—De Morgan's law, $4$.

*$\ {\sim}p\hspace{4em}$—modus tollens, $1,5$.

*$\ {\sim}p\,\wedge{\sim}r\hspace{1.5em}$—conjunction introduction, $6,3$.

