What fallacy would this be?

I was having a conversation with an atheist and he claimed that because there are hundreds of religions in the world, and they all make contradictory claims yet all claim absolute authority, therefore all religions are false.

I suppose the form would be something like: Claims X and Y both claim absolute truth, yet they are contradictory, therefore claims X and Y are both false.

It could very well be the case that the conclusion is true, but the premises used to get to the conclusion seem very fallacious.

I've tried looking throughout the book but can't seem to find it.

The atheist screwed up a common atheist talking point, which is that because there are hundreds of religions in the world, and most of them make contradictory claims while claiming absolute authority, therefore they can't all be right or we should be suspect of claims of sources of absolute authority (this point does not make the fact that one of them may be right impossible). This seems like such a obvious error that either the atheist was really not that smart or there was some misunderstanding.

I think the form of this argument would be:

Both X and Y cannot be true
Therefore, both X and Y are false

The error is in the assumption that because both of them cannot be true that one of them cannot be true. There may be a rule in formal logic like this, but I cannot think of it at the moment. I'll edit this if it comes to me.

Hmm, seems like a cocktail of fallacies: ad hoc, ad populum and Petitio Principii, false premise and false conclusion

Hundreds of religions claim absolute authority. Ad populum. ad hoc
All religions contradict each other. ad hominem (guilt by association)
Because all religions claim absolute authority and contradict each other, they are false! (Petitio Principii: assuming what needs to be proven.

X is false,
Y is false
Therefore X and Y are true ( false premise, false conclusion