I was reading over notes that explain the following:
When discussing fields, we should distinguish that which can be claimed as a basic field property ((A),(M), and (D)) from properties that can (and must) be proved from the basic field properties. For example, given that $(\mathbb{F},+,\bullet)$ is a field, we can claim that $(\forall x)(\forall y)(x,y \in \mathbb{F} \rightarrow x + y \in \mathbb{F})$ as an alternative description of property (A1) while we can not claim that additive inverses are unique
If (A1) is originally defined as follows,
(A1) $+: \mathbb{F}$ x $\mathbb{F} \rightarrow \mathbb{F}$
Could we still replace (A1) with (A1)* $(\forall x)(\forall y)(x,y \in \mathbb{F} \rightarrow x + y \in \mathbb{F})$ if we did not list out all the field axioms that $(\mathbb{F},+,e)$ satisfies (that is, if we did not know that $(\mathbb{F},+,e)$ is a field)?