Questions about first-order formulas of set theory

Question

I am new in set theory. I have 4 simple questions about first-order formulas of set theory.

1. Dose $\exists x\in Y(\varphi(x))$ stand for $\exists x(x\in Y\wedge\varphi(x))$ ?

2. Dose $\forall x\in Y(\varphi(x))$ stand for $\forall x(x\in Y\rightarrow \varphi(x))$ ?

3. $\exists x\in Y(\varphi(x))\Longleftrightarrow\neg\forall x\in Y(\neg\varphi(x)) $ holds ?

4. Is $\exists y\in Z\big[\forall x\in y\big[x\subseteq U\big]\big]$ a well-formed first-order formula ?

Answer 1

1. yes
2. yes.

3. holds whenever $Y\ne\emptyset$.

4. regarding $Z$ and $U$ as fixed parameters, yes, the formula is well-formed.