# $(A=B \iff B=C) \implies (A=C)$?

Perhaps I'm just too tired, but is it valid to say $(A=B \iff B=C) \implies (A=C)$ ? If so, what justifies it?

The thing is, by assuming just $(A=B \iff B=C)$ I don't see how we reach $A=C$, for I would also need $A=B$ or $B=C$ for hypothesis.

Thanks in advance for any clarification.

-
Hint: What happens if all the three $A,B,C$ are pairwise distinct? –  Tobias Kildetoft Sep 11 '13 at 18:39
@zerosofthezeta: $A=B,B=C$ is not the same as $A=B\iff B=C$. –  abiessu Sep 11 '13 at 18:46

## 1 Answer

No this is not true. What you might conclude is that $(A=B \implies B = C)\implies ((A=B)\implies A = C)$ and going the other way $(A=B \Longleftarrow B = C)\implies ((B=C)\implies A = C)$.

-