In Michael O'Leary's book A First Course in Mathematical Logic and Set theory he derives the intersection of a set from the Subset Axiom (Axiom Schema of Separation) and gives the following definition:
"Let $\mathcal{F}$ be a set. By a subset axiom, there exists a $C$ such that
$x\in C\leftrightarrow x\in\bigcup\mathcal{F}\wedge\forall c(c\in\mathcal{F}\rightarrow x\in c)$"
My question is about the conditional $\forall c(c\in\mathcal{F}\rightarrow x\in c)$. When read aloud this definition makes sense, however stating $\forall c(c\in\mathcal{F}\wedge x\in c)$ also makes sense. What would $C$ contain if the implication was exchanged for a conjunction?