1. ~S ∧ ~R                Premise  

2. ~S                     1, Simplification (SIMP)

3. ~R                     1, SIMP

4. ~R ∧ ~S                3,2 Conjunction Introduction (CI)

// Note, we could have just done Commutation of 1 to arrive at (~R ∧ ~S) in one step

5.:: S v R                Conditional Proof (CP) assumption #1

// Either S or R is true, let's assume S is true first...

6.:: S                    CP assumption #2 

7.:: ~(~R ∧ ~S)           Clearly (~R ∧ ~S) can not be true, because S is true.  The rule name here escapes me.

// Now assume R is true

8.:: R                    CP assumption #3

9.:: ~(~R ∧ ~S)           Clearly (~R ∧ ~S) can not be true, because R is true.  Same rule as line 7.

10.:: (S v R) --> ~(~R ∧ ~S)     5-9, CP

11.:: (~R ∧ ~S) --> ~(S v R)     10, Transposition (if A --> B then ~B --> ~A)

12. (~R ∧ ~S) --> ~(S v R)       5-11, CP

*DONE*

 

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