SOLUTION: Formal proof: In the text box below, use the proof method (M9) to construct a formal proof to demonstrate that the following argument is valid: A v (~B v ~C), A ⊃ (D ⊃ E),

Question 1188426: Formal proof: In the text box below, use the proof method (M9) to construct a formal proof to demonstrate that the following argument is valid:
A v (~B v ~C), A ⊃ (D ⊃ E), ~(~B v ~D) /.: C ⊃ E
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!

1.  A ∨ (~B ∨ ~C)
2.  A ⊃ (D ⊃ E)
3.  ~(~B ∨ ~D)     /.: C ⊃ E

               |4.  C                 Assumption for conditional proof 
               |5.  (A ∨ ~B) ∨ ~C     1, Association
               |6.  ~C ∨ (A ∨ ~B)     5, Commutation
               |7.  ~~C               4, Double negation
               |8.  A ∨ ~B            6,7  Disjunctive syllogism   
               |9.  ~~B • ~~D         3, deMorgan's law
               |10. B • D             1, Double negation
               |11. B                 10, Simplification
               |12. ~~B               11, Double negation
               |13. ~B ∨ A            8, Commutation
               |14. A                 13,12, Disjunctive syllogism
               |15, D • B             10, Commutation
               |16. D                 17, Simplification
               |17. (A • D) ⊃ E       2, Importation
               |18. A • D             14,15, Conjunction
               |19. E                 17,18, Modus ponens
20. C ⊃ E      lines 4-19    Conditional proof  

Edwin

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