(related to Lemma: Negation of an Implication)
$x=$"You work."
$y=$"You earn some money."
Then the following compound propositions are equivalent according to the lemma we have just proven:
$\neg(x\Rightarrow y)$: "It is not true that if you work, then you earn some money."
$x \wedge \neg y$: "(It is possible that) you work and you do not earn any money."
Very often, implications are wrongly negated in argumentation, causing a lot of confusion. If you ask your friends what is the opposite of "If you work, then you earn some money", you will probably get different (wrong) answers. Some of the answers you might receive could be:
"If you do not work, then you do not earn any money."
"You earn some money, even if you do not work."
That's why this lemma is not only important for mathematical proving but also to analyze if the arguments somebody confronts you with are logically correct or not.