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Suppose there are two theorems $A \Rightarrow B$ and $C \Rightarrow A$. Then we have $C \Rightarrow B$.

Now comparing $A \Rightarrow B$ and $C \Rightarrow B$, we know that $C \Rightarrow A$ means C is a stronger condition than A. Is it to say $A \Rightarrow B$ is stronger or weaker than $C \Rightarrow B$? I personally think $A \Rightarrow B$ is a stronger result/theorem than $C \Rightarrow B$, but I also saw $A \Rightarrow B$ is said to be weaker than $C \Rightarrow B$.

Thanks!

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    @user1729: I'm sort of taking $C\Rightarrow A$ as if it were an axiom (or at least, an assumption). Notice that the last line of my answer below makes that explicit, because as you say, without that assumption neither is stronger than the other.2012-07-05

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Suppose we later find that $D\Rightarrow A$. Then we can use $A\Rightarrow B$ to show $D\Rightarrow B$, but we can't use $C\Rightarrow B$ to do that.

Conversely, if we find that $E\Rightarrow C$, then we can use either $A\Rightarrow B$ or $C\Rightarrow B$ to prove $E\Rightarrow B$, since the latter can be recovered from the former and $C\Rightarrow A$.

So, in the presence of $C\Rightarrow A$, the statement $A\Rightarrow B$ is stronger than $C\Rightarrow B$.

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    I would be intrigued to hear the downvoters reason for the downvote?2012-07-05
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You are correct, $A \Rightarrow B$ is stronger than $C \Rightarrow B$ as it is the more direct path.