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If I am asked to give a direct proof of the implication "if the sky is blue, 7 is prime", would it be considered a valid direct proof to conclude 7 is prime through a chain of implications which have nothing to do with the sky being blue? (the idea being that any implication whose consequent is always true is true, regardless of whether the antecedent is true or false).

I am asking because I was taught the definition of a direct proof is a proof which assumes the antecedent, and through a chain of implications concludes the consequent- however, I am not sure if the chain of implications have to be built off of the antecedent, or if they can have no relationship to the antecedent, in order for this proof to be considered a direct proof. So would my proof of the implication I gave constitute a valid direct proof?

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    $\begingroup$ It should be easy to see that, in most cases, there isn't a way to prove that in the first place. "If $x$ is even, then $x+1$ is odd" would be an example: you can't prove that $x$ is even because you don't even know what $x$ is. Of course, whether the assumption is ever valid is a concern in its own right, especially when in a piece of a grander puzzle - but in principle, to prove $P \implies Q$, you will assume that $P$ is true and work from there to deduce $Q$. $\endgroup$
    – PrincessEev
    Jun 5 at 1:07

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A direct proof is simply a proof whose assumptions are drawn from just the set of axioms, established theorems and other givens. This is not to say that a direct proof (actively) assumes any particular member of this set!

In particular, given that the sky is blue, a direct proof of 7 being prime does not preclude ignoring that given.

Proving the statement "if A, then B" by just proving B is a valid direct proof as long as A is indeed a meaningful statement.

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