Conditional fallacy

A conditional fallacy may or may not be a logical fallacy, conditional on whether or not one of its premises is accepted.

Conditional fallacies are almost always informal fallacies. However, if comparing multiple systems of logic (which would imply different valid and invalid logical structures, and thus different formal fallacies) then it may be possible for a formal fallacy to be conditional.

Form
An informal conditional fallacy follows the form:

Non-fallacious
This is not always fallacious. For example, a good (rational) conditional argument:

Fallacious
The problem only occurs when Y is something stupid. For example, a bad (fallacious) argument from authority: