Equality is required to be defined under L, and the only rules required for L extended by the epsilon operator are modus ponens and the substitution of A(t) to replace A(x) for any term t.
For equality to be valid within L, explicit definitions must be given, and when L is extended through the use of the epsilon operator, it is subject to two key rules: modus ponens and the replacement of A(t) with A(x) in any term or expression.