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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.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
This property follows directly from applying the chord theorem to a third chord (a diameter) going through S and the circle's center M (see drawing). The theorem can be proven using similar triangles (via the inscribed-angle theorem ).
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 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 z which satisfy the property .
Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as ( a + b ) = ( b + a ) {\displaystyle (a+b)=(b+a)} .