The logical axioms of a first-order language $\mathscr{L}$ are divided into three groups, and they are as follows:
Sentential axioms: every formula $\varphi$ of $\mathscr{L}$ which can be obtained from a tautology $\psi$ of $\mathscr{S}$, where $\mathscr{S}$ is a set of statements which are formulas of $\mathscr{L}$, is a logical axiom of $\mathscr{L}$. We will call $\varphi$ a tautology of $\mathscr{L}$ if this is the case.
The quantifier axioms are
Identity axioms: if $x,y$ are variables, $t(v_0\cdots v_n)$ is a term, and $\varphi(v_0\cdots v_n)$ is an atomic formula, then the formulas are logical axioms.
In addition to the logical axioms, there are two rules of inference:
Rule of Detachment or modus ponens: from $\varphi$ and $\varphi\rightarrow\psi$ we may infer $\psi$.
Rule of Generalization: from $\varphi$ we may infer $(\forall x)\varphi$.