Everything is equal

Question

This bothers me for a while:

Proof $a = b$ $$a=b$$ $$b=a$$ $$a+b=b+a$$ $$a+b=a+b$$ $$0 = 0$$

This looks like a proof for $a=b$, but it shouldn't work like this. But i can't put my finger on why it's wrong.

Answer 1

This merely proves that if $a=b$ then $0=0$. Try proving it the other way around and you'll see that you can't reverse the steps.

Answer 2

There's nothing wrong with this. You started by assuming that $a=b$, so your proof only holds if, indeed, $a$ is equal to $b$.

Answer 3

If you reverse the steps, there is no viable rule or theory that lets you go from $$a+b=b+a$$ to $$b=a$$

Answer 4

Make the proof complete by adding either $\;\Rightarrow\;$, $\;\Leftarrow\;$, or $\;\Leftrightarrow\;$ inbetween each two expressions. Then you will see that you have proved $\;a = b \;\Rightarrow\; 0 = 0\;$.

Answer 5

It starts off with what it attempts to prove.

Prove $a=b$, is not supposed to start off with $\text{Let }a=b$.