Is there any branch of mathematics in which it is false that If A = B, then B = A ? ?

The relation "is equal to" is the canonical example of a binary equivalence relation which is reflexive, symmetric and transitive.

If the equal sign (=) is being used to represent the "is equal to" relation, then it is always true that:

A = A (reflexive)

A = B → B = A (symmetric)

A = B, B = C → A = C (transitive)

But it is possible to redefine the meaning of the equal sign to where this wouldn't be true. So how are you defining '='?

Statement A = all dogs are 4 legged creatures.

Statement B = All 4 legged creatures are dogs.

B is not equal to A but A = B.

Logic could be such branch.

But if A and B are numbers, then it can't be false. 

All boy children are male.

All males are not boy children.

I do not know everything ... anything is possible...