Open axioms of equality I have a doubt. I need help.
Can the basic axioms of equality be presented as "open axioms"?
I) (reflexivity)
$\qquad x = x$            
II)  (Substitutivity)
$\qquad (x = y) \to \big(F (x, x) \to F (x, y)\big)$ 
 A: Yes, we can.
Consider the Natural Deduction proof system.
We can derive $(x=x)$ from the "closed" version: $\forall x \ (x=x)$ simply using $\forall$-elim rule.
Conversly, from the open axiom $\vdash (x=x)$, we can derive $\vdash \forall x \ (x=x)$ using $\forall$-intro.
There are no assumptions (i.e. $\Gamma = \emptyset$) and thus the proviso of the rule: "$x$ not free in $\Gamma$", is satisfied.

The open equality axiom $(x=x)$ is obviously valid.
See e.g.: 


*

*Dirk van Dalen, Logic and Structure, Springer (5th ed. 2013), page 67.


The definition of true in a structure $\mathfrak A$ is restricted to sentences, i.e. "closed" formulas.
For open ones, it adopts the convention that:

$\mathfrak A \vDash \varphi \ $  iff $ \ \mathfrak A \vDash \text {Cl}
 (\varphi)$,

where $\text {Cl} (\varphi)$ is the universal closure of $\varphi$. 
A different approach is adopted by:


*

*Herbert Enderton, A Mathematical Introduction to Logic, Academic Press (2nd ed. 2001), page 83.


In this case, the meaning and the truth-value of an open formula with respect to an interpretation $\mathfrak A$ is defined for specific "instances" of the formula, obtained through the variable assignment device.
The basic notion is that of satisfaction of a formula $\varphi$ by an assignment $s$ in a interpretation $\mathfrak A$ (in symbols: $\mathfrak A, s \vDash \varphi$).
A formula $\varphi$ is true in $\mathfrak A$ when it is satisfied by every assigment.
Consequently, $(x=x)$ is true in every interpretation.
A: I agree with Mauro ALLEGRANZA. 
The use of open instead of closed formulas as axioms comes from Hilbert proof systems which have a rule of inference Universal Generalization statting that $\forall x F(x)$ can be derived from $F(x)$. The rule $\forall$ introduction of Gentzen's Natural Deduction is similar to Universal Generalization. 
David Hilbert and Wilhelm Ackermann: Grundzüge der Theoretischen Logik, Springer, Berlin, Germany, 1928 (An English translation was published in 1950, see below) 
David Hilbert and Wilhelm Ackermann: Principles of Mathematical Logic, AMS Chelsea Publishing, Providence, Rhode Island, USA, 1950 
