Suppose I want to prove $\sqrt{7}$ is not a rational number. I suppose it is and it brings me to a contradiction. Here how it goes line be line:
First line. $\sqrt{7}=\frac{m}{n}$
Second line (after squaring both sides of the equality above). $7n^2=m^2$
Third line. A solution for this equality contradicts the uniqueness part of the fundamental theorem of arithmetics, thus it has no solution.
I used the argument above in my number theory class today. It was hard for one of the students to accept it for an interesting reason I had never seen before.
No solution for $7n^2=m^2$ means $7n^2$ is not a square number, but no solution for $\sqrt{7}=\frac{m}{n}$ means $\sqrt{7}$ is not a rational number. These two statements are not the same, are they? Asked the student!
The lecture that happened to be me had not an answer, or at least, at that moment, and even now, could not find a similar chain of algebraically equivalent statements that each one brings a different meaning to mind, while it is "easy" to see those meanings refer to the same thing.
I guess my question is do you know an example like the one that I tried to summarize in the title and explain in the body of the post. Or, should I ask, if you were the lecturer what was your answer to the student?