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The substitution property of equality, or Leibniz's Law (though the latter term is usually reserved for philosophical contexts), generally states that, if two things are equal, then any property of one, must be a property of the other
In logic, equality is a primitive predicate (a statement that may have free variables) with the reflexive property (called the Law of identity), and the substitution property. From those, one can derive the rest of the properties usually needed for equality.
The most common convention, known as first-order logic with equality, includes the equality symbol as a primitive logical symbol which is always interpreted as the real equality relation between members of the domain of discourse, such that the "two" given members are the same member. This approach also adds certain axioms about equality to the ...
To guarantee a sequent's validity, however, we must conform to carefully specified rules. The rules can be divided into four groups: the propositional rules (1-10), the predicate rules (11-14), the rules of equality (15-16), and the rule of substitution (17). Adding these groups in order allows one to build a propositional calculus, then a ...
There are a few other reasons to restrict study of first-order logic to normal models. First, it is known that any first-order interpretation in which equality is interpreted by an equivalence relation and satisfies the substitution axioms for equality can be cut down to an elementarily equivalent interpretation on a subset of the original ...
indicates that the column's property is always true for the row's term (at the very left), while indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and in the ...
Substitution, written M[x := N], is the process of replacing all free occurrences of the variable x in the expression M with expression N. Substitution on terms of the lambda calculus is defined by recursion on the structure of terms, as follows (note: x and y are only variables while M and N are any lambda expression): x[x := N] = N
Once substitution has been made explicit, however, the basic properties of substitution change from being semantic to syntactic properties. One most important example is the "substitution lemma", which with the notation of λx becomes (M x:=N ) y:=P = (M y:=P ) x:=(N y:=P ) (where x≠y and x not free in P)