Mistakes regarding proofs

I had a teacher who would repeat many times in different classes "If you write 0 = 0 I'm going to give you a zero in the exam!".

Suppose we want to prove that [math]\displaystyle{ a + b = c + d }[/math]. We begin by saying that [math]\displaystyle{ a + b = x }[/math]. Then we say that [math]\displaystyle{ c + d = x }[/math]. Therefore [math]\displaystyle{ x = x \iff a + b = c + d }[/math]. Eureka! We didn't prove anything! What is wrong in the reasoning that we just did? The mistake is that we assumed that the equation is true without knowing whether it holds or not. We can't do such assumptions!