On logical connector implies It may appear at first glance that this question had been asked over and over here. But I feel that the question that is in my mind is slightly different from what has already been asked. Here it is:

What would have happened if $P \implies Q$ was taken false when $P$ was false and $Q$ was true?

I read several answers for the question: "Why false implies True is false?". I could gather some information from the answers. One is that irrespective of the truth value of $Q$, if $P$ is false, we treat $P \implies Q$ to be true vacuously. But this didn't answer my question, what would have happened if I had taken it to be false?
Another answer, which though addressed my question, I wasn't much satisfied with. This was that answer: If we had taken false implies true to be false, then truth tables of $\implies$ and $\leftrightarrow$  would have been one and the same.
Any answer in the direction of consequence of false implies true false is highly appreciated. I mean, would there have been any logical fallacy, contradiction or paradox of so kind if I had assumed false implies true to be false?


What would have happened if $P \implies Q$ was taken false when $P$ was false and $Q$ was true?

To avoid inconsistencies, at the very least, you would have to invalidate at least one step in the following proof that $\neg P \implies [P \implies Q]:\\$

(Screenshot from my proof checker)
You might, for example, have to disallow or somehow restrict proof by contradiction (line 5), or elimination of '$\neg\neg$' (line 6).
