THEOREM

*******

 

     ~A => [A => B]  (Truth Table, lines 3-4)

 

PROOF

*****

 

    Suppose...

 

      1     ~A

            Premise

 

         Suppose...

 

            2     A

                  Premise

 

             Suppose to the contrary...

 

                  3     ~B

                        Premise

 

             Obtain contradiction...

 

                  4     A & ~A

                        Join, 2, 1

 

         Apply Conclusion Rule (By Contradiction)

 

            5     ~~B

                  4 Conclusion, 3

 

Apply Double Negation Rule

 

            6     B

                  Rem DNeg, 5

 

    Apply Conclusion Rule (Direct Proof)

 

      7     A => B

            4 Conclusion, 2

 

Apply Conclusion Rule (Direct Proof)

 

As Required:

 

8     ~A => [A => B]

      4 Conclusion, 1

 

 

 

 

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