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A substitution is a syntactic transformation on formal expressions. ... and the substitution property of equality, [5] also called Leibniz's Law. ...
The first two are given by the substitution property of equality from first-order logic; the last is a new axiom of the theory. Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by Azriel Lévy .
Substitution for formulas. For any variables x and y and any formula φ(z) with a free variable z, then: x = y → (φ(x) → φ(y)). These are axiom schemas, each of which specifies an infinite set of axioms. The third schema is known as Leibniz's law, "the principle of substitutivity", "the indiscernibility of identicals", or "the replacement ...
The converse of this axiom follows from the substitution property of equality. 2) Axiom Schema of Specification (or Separation or Restricted Comprehension ): If z is a set and ϕ {\displaystyle \phi } is any property which may be satisfied by all, some, or no elements of z , then there exists a subset y of z containing just those elements x in ...
Independence of irrelevant alternatives (IIA) is an axiom of decision theory which codifies the intuition that a choice between and should not depend on the quality of a third, unrelated outcome .
The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism ϕ : R → S {\displaystyle \phi :R\to S} and an element x in S there exists a unique ring homomorphism ϕ ¯ : R [ t ] → S {\displaystyle {\overline {\phi }}:R[t]\to S} such that ϕ ¯ ( t ) = x {\displaystyle ...
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Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation or integration (integration by substitution). A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial: