With any two contradictory premises you can prove any unrelated third premise. Example:
If 1) this post was written in Iceland, and 2) this post was not written in Iceland, then 3) blue whales are fluent in Swahili.
For it to be false that the third premise follows from the first two, there must exists at least one case where the first two premises are true but the third one is not. Included in that requirement is therefore the requirement that there must be at least one case where the first two premises are true. Since they are contradictory, this cannot occur, and thus cannot be a case where the first two are true and the third is false. Thus the third premise follows from the first two.
You can do weird things with logic like that. One of my favorites is, "In any bar where there are people who may or may not be drinking, there will always exist at least one person who, if he is drinking, everyone else in the bar is drinking".
Which sounds ridiculous. But you have two possibilities: either everyone is drinking, or there's at least one person who's not drinking. If everyone is drinking, then your "one person" can be any of them. If not everyone is drinking, you can pick any one person who isn't drinking to be your "one person". Since the statement has a premise "If he is drinking" - that premise immediately fails, and so there is no requirement in the rules of logic that the second part "everyone else in the bar is drinking" must hold.
It's a logically consistent statement, but breaks the expectations of conversational language. A reader naturally wants to interpret it as meaning that there's going to be a person who, if he suddenly decides to start drinking, then everyone in the bar is going to suddenly pick up a glass and start drinking. But the statement in its basest form doesn't claim that, it is only a description of a moment of time and requires no consistency on who does what between different time periods.