I think that all logical connectives, quantifiers and identity are derivable from a simple semi-formal inference rule denoted by "|" to represent "infers" and this is not to be confused with the Sheffer stroke nor any known logical connective.
A| C can be taken to mean the "negation of A"
A,B| A can be taken to mean the "conjunction of A and B"
x| phi(x) can be taken to mean: for all x. phi(x)
x, phi(y)| phi(x) can be taken to mean: x=y
The idea is that with the first case we an arbitrary proposition C is inferred from A, this can only be always true if A was False, otherwise we cannot infer an "arbitrary" proposition from it.
Similarly with the second case A to be inferred from A,B then both of those must be true.
Also with the third condition to infer that for some constant predicate phi it is true that given x we infer phi(x) only happens if phi(x) is true for All x.
With the fourth case for an 'arbitrary' predicate phi if phi(y) is true and given x we infer that phi(x) is true, then x must be identical to y.
Anyhow the above kind of inference is somewhat vague really, it needs to be further scrutinized.