SOLUTION: find the proof of the following: P & Q, P -> R |- R & Q (P v Q) -> R |- ~R-> ~Q

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Question 1109316: find the proof of the following:
P & Q, P -> R |- R & Q
(P v Q) -> R |- ~R-> ~Q
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

1.  P & Q 
2.  P -> R     / R & Q

3.  P            1, simplification
4.  R          2,3, modus ponens
5.  Q & P        1, commutation
6.  Q            5, simplification
7.  R & Q      4,6, conjunction  

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1.  (P v Q) -> R    /  ~R-> ~Q

2.  ~R -> ~(P v Q)  1, transposition
3.  ~R -> ~P & ~Q   2, deMorgan's law
4.  ~R -> ~Q & ~P   3, commutation   
5.  ~R -> ~Q        4, simplification

Edwin