THEOREM

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From a falsehood, all things follow.

 

A => [~A => B]

 

Latin: Ex falso quodlibet. (ETQ)

 

See: The Principle of Explosion, Wikipedia

 

 

PROOF

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    Suppose...

 

      1     A

            Premise

 

         Suppose...

 

            2     ~A

                  Premise

 

             Suppose to the contrary...

 

                  3     ~B

                        Premise

 

             Obtain the contradiction...

 

                  4     A & ~A

                        Join, 1, 2

 

         Apply Conclusion Rule (By Contradiction)

 

            5     ~~B

                  4 Conclusion, 3

 

            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