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The rules of formation for the language of set theory are the following:

  1. If $a$ is a variable and $b$ is a variabe, then $a\,{\in}\,b$ is a WFF.

  2. If $P$ is a WFF and $Q$ is a WFF, then ${\lnot}P$,$P{\land}Q$,$P{\lor}Q$,$P{\implies}Q$,$P{\iff}Q$ are all WFFs.

  3. If $a$ is a variable and $P$ is a WFF, then ${\forall}a(P)$ and ${\exists}a:(P)$ are WFFs.

Although point 1. is easy to understand, I have problems regarding 2. and 3. Why do things like $P{\iff}Q$ need to be defined as WFFs? Considering that $P{\iff}Q$ means that $(P{\implies}Q){\land}(Q{\implies}P)$, isn't it enough to just say that $P{\land}Q$ and $P{\implies}Q$ are WFFs and not mention $P{\iff}Q$ at all?

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We do not define formulae but symbols.

Having said that: YES, it is possible to "save" conncetives using only, e.g. $\lnot$ and $\land$ as primitive and introduce the others: $\lor, \to, \leftrightarrow$ as abbreviations through the usual definitions.

In the same way, we can use only $\forall$ as primitive and define $\exists$ as $\lnot \forall \lnot$.

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