The fundamental nature of algebra rests on the basic rule that whenever two numbers, variables, or expressions are equal, either one can be replaced at any time by the other one. For example, if we assume that a = b implies that b = a (the symmetric property of equality), then if I let a = 7, we have 7 = b implies b = 7. Without the basic rule, there is no way to apply the symmetric property to actual numbers.
The transitive property of equality states that if a = b and b = c, then a = c. This seems to be just a specific case of the basic rule, substituting a for b in the second assumption.
So if none of the standard axioms can be understood without the basic rule, how is transtive property a meaningful axiom?
Let's assume the transitive property of equality is not true. Thus, we cannot assume that because a=b, a+c=b+c. Why? The transitive property of equality says that if x=y and y=z then x=z, so adding the same value to each side of an equation would have no meaning, as they would be not equal.