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