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Proof

(related to Lemma: Mixing-up the Sufficient and Necessary Conditions)

We want to prove that the mixing-up the sufficient and necessary conditions is a fallacy. * The fallacy can be formulated in propositional logic as $(p\Rightarrow q)\Rightarrow (q\Rightarrow p).$ * The definition of implication "$\Rightarrow$" gives us the following truth table of the function:

$[[p]]_I$	$[[q]]_I$	$[[p\Rightarrow q]]_I$	$[[(q\Rightarrow p)]]_I$
$0$	$0$	$1$	$1$
$0$	$1$	$1$	$0$
$1$	$0$	$0$	$1$
$1$	$1$	$1$	$1$

In the second row, the premise $p\Rightarrow q$ is true, while the conclusion $q\Rightarrow p$ is false.
This is, by definition, the criterion for a fallacy.
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References

Bibliography

Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016