How to prove $C$ from $A \leftrightarrow (B \leftrightarrow C)$ and $A \leftrightarrow B$?

Question

How does one prove $C$ from the premises: $A \leftrightarrow (B \leftrightarrow C)$ and $A \leftrightarrow B$ ?

I've tried to prove $C$ by contradiction, using a sub-proof which presumes $\neg C $, but although I can conclude all of the following in the subproof: $\neg A$, $ \neg B$, $ \neg (B \leftrightarrow C)$, I'm unable to find a contradiction this way.

I've been stuck on this for the whole day, and I think I might be over-thinking the problem.

Note: I want to prove this using the basic first-order logic rules (I'm using the First-Order Logic from the Language, Proof and Logic book).

Answer 1

Due to the transitivity of $\leftrightarrow$ and due to the fact that $A$ comes up on both premises 'at the same level', I find it natural to focus on $A$ and let it act as a pivot of sorts.

Start by proving $A\lor \neg A$ and perform $\lor$-$\text{Elim}$ on this disjunction.

In the first case just use $\leftrightarrow$-$\text{Elim}$ successively on the premises to get $C$.

In the second case (where one starts a subproof with the premise $\neg A$), use the premise $A\leftrightarrow B$ to get $\neg B$ and the premise $A\leftrightarrow (B \leftrightarrow C)$ to get $\neg(B\leftrightarrow C)$ (in both cases by negation introduction).
Now assume $\neg C$, prove $\neg B\leftrightarrow \neg C$ and from this last statement get $B\leftrightarrow C$.
At this point you can find a contradiction allowing you to conclude $C$ in the subproof whose premise is $\neg A$.

I leave the proof below.

Answer 2

This basically follows from the associativity of $\leftrightarrow$. But let's pretend that we didn't know that.

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We consider two exhaustive, mutually exclusive cases.

Case 1: Suppose that $B$ is true. Then since $A \leftrightarrow B$ is true, we know that $A$ is true. Thus, since $A \leftrightarrow (B \leftrightarrow C)$ is true, we know that $B \leftrightarrow C$ is true. But then since $B$ is true, we know that $C$ is true, as desired.

Case 2: Suppose that $B$ is false. Then since $A \leftrightarrow B$ is true, we know that $A$ is false. Thus, since $A \leftrightarrow (B \leftrightarrow C)$ is true, we know that $B \leftrightarrow C$ is false. But then since $B$ is false, we know that $C$ is true (otherwise, if $C$ was actually false, then $B \leftrightarrow C$ would be true, a contradiction). So we're done!

Answer 3

I would use the algebraic machinery:

$A\leftrightarrow B\quad$ iff $\quad 1\oplus A\oplus B$

and get $(A\leftrightarrow (B\leftrightarrow C))\wedge (A\leftrightarrow B)\quad$ iff $\quad (1\oplus A\oplus 1\oplus B\oplus C)(1\oplus A\oplus B)=$ $=(A\oplus B\oplus C)(1\oplus A\oplus B)$ $=A\oplus B\oplus C\oplus A\oplus AB\oplus AC\oplus AB\oplus B\oplus BC=$ $=C\oplus AC\oplus BC=C(1\oplus A\oplus B)\quad$ iff $\quad C\wedge(A\leftrightarrow B)$, which imply C.

Which considering the note may not be as relevant, but...

Answer 4

$ \newcommand{\calc}{\begin{align} \quad &} \newcommand{\calcop}[2]{\\ #1 \quad & \quad \text{"#2"} \\ \quad & } \newcommand{\endcalc}{\end{align}} $Your proof by contradiction approach is fine, here is how you can complete your proof.

Assume $\;C\;$ is false, then $$\calc A \leftrightarrow (B \leftrightarrow C) \calcop{\leftrightarrow}{using what we know about $\;C\;$} A \leftrightarrow (B \leftrightarrow \text{false}) \calcop{\leftrightarrow}{left hand side: use $\;A \leftrightarrow B\;$; right hand side: simplify} B \leftrightarrow \lnot B \calcop{\leftrightarrow}{logic} \text{false} \endcalc$$

which is a contradiction. Therefore $\;C\;$ is true.