With all due respect, get your head out of your arse and read this from what I posted:
While modus ponens is one of the most commonly used argument forms in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the “rule of definition” and the “rule of substitution”.
Emphasis is mine. I cannot scream hard enough to get this simple message across to your flipping head. You are reading it wrong, and if you had done one class of prepositional calculus you would have known, therefore you haven’t.
As for your foundationalist pursuits, most of science advances without getting back to the foundations, just as calculus was in practical use long before it was formally proven. So you see a person (OP) struggling with basic conception and composition of his argument, let alone the formal expression, and you raise the bar to the level of logical foundations of mathematics? If not dishonest, this is utterly unproductive.
No one is attempting to prove bleeding P->Q here.
Sure, and when ~P^Q, then P->Q is still not false, and you can further use it in a proof, in the context of other given statements.
This was never presented as a method to show that P->Q, which arguably can only be shown with data.