I was just trying to abstractly capture a proof technique, and here is my little doubt as follows.
Suppose we want to prove $A \implies B$
We already have $A \implies A_{f}$
Now, of course we are to prove B yet, but still on a rough work level, given $A$, suppose $B$ holds, then we are able to deduce that
$A \land B \implies C$
Now consider that we can indeed prove that $C$ holds, independently of $B$, i.e. by $A \implies C$,
Then,
$((A \implies A_{f}) \land (A \implies C)) \implies (A \implies (A_{f} \land C))\;$
(Indeed both sides are logically equivalent as truth tables are same.)
Now we have that $A \implies (A_{f} \land C)$. In the setting, it is also provable that $(A_{f} \land C) \implies B$. This suffices to prove that $A \implies B$, But my main question is that "does the derivation of $C$ from $A \land B \implies C$ have any effect on the validity of the proof?"
$C$ was obtained as a rough work from assuming $B$, along with the given $A$, and that was it. I know that one shouldn't assume what one's trying to prove, but for the case of $C$ here, its truth was supplied by $A$ and not $B$, as it was provable by $A \implies C$
$C$, for the matter of proof, was deducible from $A$ alone, and the two truths deduced from $A$ alone suffice to prove $A \implies B$.
The rough work of $A \land B \implies C$ just served as a guess for looking out for such $C$ which could aid in proof.
It might've not been much of a question, but I just wanted to confirm a doubt which I hadn't encountered before.
