to mean that and are names for the same object. I will not make a distinction between an object and its name.

Let be a proposition involving the object . Let be a proposition obtained
by replacing any or all occurrences of in by . Then
. We call
this property of equality the *substitution property*.

and

and

We will frequently make statements like

The justification for this is

Hence, if , then by the substitution property,

and

are both true, the result of substituting the in the second equation by yields

which is false.

The proper conclusion that follows from (1.40) and (1.41) is

(The use of parentheses is discussed in Remark 2.50.)

as an abbreviation for

If (1.43) is true, then by several applications of transitivity, we conclude that