Question 1210240
<pre>
We don't have access to your material, so we have no idea what your material
creator meant by proof method "M9".  I'll just prove whatever method works best,
You can put arrows before line numbers.  I don't do that.  We don't know
anything about "Show more buttons".
 
~(T v U), S, R ≡ ~S   /.: ~(U v R)

1.  ~(T v U)
2.   S
3.   R ≡ ~S   /.: ~(U v R)
4.  ~T • ~U                 1, DeMorgan's law
5.  ~U • ~T                 4, Commutation
6.  ~U                      5, Simplification
7.  (R ⊃ ~S) • (~S ⊃ R)    3, Material equivalence
8.  R ⊃ ~S                 7, Simplification
 9.  ~~S ⊃ ~R               8, Transposition
10.  S ⊃ ~R                 9, Double negation
11.  ~R                 10, 2, Modus tollens
12.  ~U • ~R            6, 11, Conjunction
13.  ~(U v R)              12, DeMorgan's law              

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1.  S v (~R • T)
2.  R ⊃ ~S         /.: ~R
                |3.  ~~R   Assumption for Indirect Proof
                |4.  R           3, Double Negation
                |5.  ~S        2,4, Modus Ponens
                |6.  ~R • T    1,5, Disjunctive Syllogism
                |7.  ~R          6, Simplification
                |8.  R • ~R    4,7, Conjunction
9. ~R         Lines 3-8 for Indirect proof.

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1.  S v (T ⊃ R)
2.  S ⊃ T
3.  ~(T ⊃ R)              /.: T
                |4.  ~T              Assumption for Indirect Proof
                |5.  ~T ⊃ ~S               2, Transposition
                |6.  ~S                  5,4, Modus ponens
                |7.  T ⊃ R               1,6, Disjunctive Syllogism
                |8.  (T ⊃ R) • ~(T ⊃ R)  7,3, Conjunction
9.  T         Lines 4-8 for Indirect Proof.

Edwin</pre>