Is this proof correct: p → q, ¬p → r, ¬q → ¬r ⊢ q

Question

Is this proof correct?

p → q, ¬p → r, ¬q → ¬r ⊢ q

1) p → q  premise
2) ¬p → r premise
3) ¬q → ¬r premise
4) -q       assumption
5) -p       MT 1,4
6) p       assumption
7) ⊥       ¬e 5,6
8) ⊥        ¬e 2,3
9 q         ¬e 8

Answer 1

This is a natural deduction proof:

$$\begin{array} {l} % p \implies q \\ % \lnot p \implies r \\ % \lnot q \implies \lnot r \\ % p \lor \lnot p \\ % \begin{array} {l|l} \quad & p \\ \hline & q \end{array} \\ % \begin{array} {l|l} \quad & \lnot p \\ \hline & r \\ & q \lor \lnot q \\ % & \begin{array} {l|l} \quad & q \\ \hline & q \end{array} \\ % & \begin{array} {l|l} \quad & \lnot q \\ \hline & \lnot r \\ & q \end{array} \\ % & q \\ % \end{array} \\ % q \\ % \end{array}$$

I left out the explanation of each step, such as Law of Exluded Middle, Or Elimination, Implication Elimination, for you to work out. And because it is really hard to format them.