I recently learned the correct ways of using the universal and existential quantifiers in the context of being in a set and exhibiting a particular property...specifically:
$\forall x \in M, \phi (x) \iff \forall x \big( x \in M \rightarrow \phi(x) \big)$
$\exists x \in M, \phi(x) \iff \exists x \big (x \in M \land \phi(x) \big )$
However, I also learned that, although it would rarely have any purposeful usage, the statement $\exists x \big (x \in M \rightarrow \phi(x) \big )$ is valid.
My question is as follows:
Is the first order symbolic statement $\forall x (x \in M \land \phi(x) )$ always false?
My thinking is that it depends on the axiomatic framework you are working in. The only time this statement could be true is if one's model permits a universal set (I think). Therefore, in Zermelo-Fraenkel Set Theory, where universal sets are disallowed, this statement is always false.
Is this the correct thinking?