The material implication operator $\to$ is not associative, i.e. $$((((A \to B) \to C) \to \cdots )\to X) \neq (A \to (B \to (C \to (\cdots \to X)))$$ I am trying to understand their difference.
What I can prove is that $$(A \to (B \to (C \to (\cdots \to X))) = (A \land B \land C\land \cdots) \to X$$
and this can be motivated by observing how for each letter $A,B,C,...$ I added as a true statement, the conclusion move one step inner into the chain eventually terminate at $X$. However, I have trouble figuring out an easier way to understand what the left associative implication chain is doing.
Attempt to use a similar trick give me confusing results
$A$: Does nothing except allowing $X$ to de-nest by one level
$A \land (A\to B)$ which is logically equivalent to $A \land B$: Does nothing except de-nest $X$ by two levels and de-nest $W$ by one level
(By de-nesting, I mean the following observation:
C
If A:
If B:
If C:
D
C
C -> D
D
B -> D
Thus because C is true, D can move up one level of context, hence de-nested by one level.)
How does the left implication chain behave as we add letters $A,B,...$ and how to rewrite it so that as many $\to$ becomes $\land$?