Proving a theorem by proving a related one

Question

I have a question that might be stupid, but it bugs me for some time. It's quite simple:

Let's say we have a conjecture, such that if the conjecture is true then another theorem will be true. Does proving the latter theorem without relying on the initial conjecture at all imply that the conjecture is also true?

As an example, it is quite known that if Riemann Hypothesis is true then many theorems will hold true. (I read that somewhere, but I don't have a source right now.) Does proving one of those theorems without using in the proof any connection with the conjecture make the conjecture to be also true?

Put in other words, let's say if R-H is true then a equals b. Would proving a equals b (without touching R-H) transition the proof to R-H?

I would suspect that others thought of this thing too, so a link to that would suffice.

Answer 1

No, it doesn't.

Say $A$ implies $B$. If $B$ is true, that doesn't imply that $A$ is true.

Consider the statement "If it rains, then I will use an umbrella".

I might be someone who uses an umbrella regardless of the weather condition. Observing that I use an umbrella doesn't mean it is raining; I might be using it as a sunshield.

Answer 2

Siong Thye Goh's answer demonstrates the flaw in the logic. I just want to add one comment.

When trying to think about the logic of proofs, don't reach for some big theorem or conjecture, or the thing you're trying to prove, as an example. Try the logic out on the simplest possible example you can think of. For example:

Suppose $(x=4)$ proves ($x$ is even). If I prove $x$ is even, does that prove $x=4$?

Since you already know that $x=4$ would make $x$ even, you don't need to be distracted by it and are free to focus just on the logic—which then becomes crystal clear.

Of course I could have chosen one of the many other theorems that would hold if $x=4$, such as "$x$ is a square", "$x$ is indivisible by $5$" . . . See what I mean? It's just easier to think about $x=4$ than the Riemann Hypothesis.