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
1. If $\varphi$ and $\psi$ are formulas of $\mathscr{L}$ and $v$ is not free in $\varphi$, then the formula $(\forall v)(\varphi\rightarrow\psi)\rightarrow (\varphi\rightarrow (\forall v)\psi)$ is a logical axiom.
2. If $\varphi,\psi$ are formulas and $\psi$ is the formula obtained by substituting each free occurrence of $v$ in $\varphi$ by a term $t$ (which contains no variables that occur bound in any place where it is substituted), then the formula $(\forall v)\varphi\rightarrow\psi$ is a logical axiom.
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 $x\equiv x$ $x\equiv y\rightarrow t(v_0\cdots v_{i-1}xv_{i+1}\cdots v_n)\equiv t(v_0\cdots v_{i-1}yv_{i+1}\cdots v_n)$ $x\equiv y \rightarrow \big( \varphi(v_0\cdots v_{i-1}xv_{i+1}\cdots v_n)\rightarrow \varphi(v_0\cdots v_{i-1}yv_{i+1}\cdots v_n)\big)$ 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 $(\forall x)\varphi$.

model-theory

first-order-logic