Contraposition
More rigorous proof of the equivalence of contrapositives

Logical equivalence between two propositions means that they are true together or false together. To prove that contrapositives are logically equivalent, we need to understand when material implication is true or false.

This is only false when P {\displaystyle P} is true and Q {\displaystyle Q} is false. Therefore, we can reduce this proposition to the statement "False when P {\displaystyle P} and not- Q {\displaystyle Q} " (i.e. "True when it is not the case that P {\displaystyle P} and not- Q {\displaystyle Q} "):

The elements of a conjunction can be reversed with no effect (by commutativity):

We define R {\displaystyle R} as equal to " ¬ Q {\displaystyle \neg Q} ", and S {\displaystyle S} as equal to ¬ P {\displaystyle \neg P} (from this, ¬ S {\displaystyle \neg S} is equal to ¬ ¬ P {\displaystyle \neg \neg P}, which is equal to just P {\displaystyle P}):

This reads "It is not the case that (R is true and S is false)", which is the definition of a material conditional. We can then make this substitution:

By reverting R and S back into P {\displaystyle P} and Q {\displaystyle Q}, we then obtain the desired contrapositive:
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