If statement A requires B to be true, is it possible to prove A without using B?

Question

[it is totally described in the title in fact]

I have a statement A that is true only if statement B is true. Is it possible to prove A without referring to B in any way? (I mean, if you use something that implies B but you d, that's not the thing)

Example (geometric):

• given some info and that point M is the center of AB, you need to prove that some two angles are equal;
• you know that if M is not the center of AB, then the angles are not equal;
• is it possible to prove that the angles are equal without using the fact that M is the center of AB?

(just an example that led me to this question)

Answer 1

Logically "$A$ requires $B$" merely rules out the combination ($A$ true, $B$ untrue).

Suppose $A$ is "$n=2$" and $B$ is "$n$ is even". Clearly $A$ requires $B$, since if $n$ is odd it can't equal $2$. But we might have proved $n=2$ from, say

• $n$ is an integer such that $1<n<e$, or
• $n=\sum_{r=0}^\infty (\frac{1}{2^r})$,

or any number of ways not involving the concept of evenness.

If $A$ requires $B$ then proving $B$ false would disprove $A$—but this doesn't mean that a proof of $A$ must use $B$.

"$A$ requires $B$" and "Proving $A$ entails proving $B$" are different propositions.