Contraposition of an implication with quantifiers I am trying to prove a theorem, and a method is by using contraposition. What is the contraposition of the phrase: $\exists x$ satisfying P $\Rightarrow$ $\forall y$ satisfy P I thought it as $\exists y$ that does not satisfy P $\Rightarrow$ $\forall x$ do not satisfy P. Is this correct?

You are correct. $\forall$ turns into $\exists$ and each logical expression gets negated.