Logic – Example of mathematical reasoning, example of dilemma?

A dilemma is a reasoning either of form

(1) if A then B
(2) if C then D
(3) A or C
(4) So, B or D

or form

(1) if A then B
(2) if C then D
(3) not B or not D
(4) So not A or not C

Are there situations in mathematics where a conclusion of the form (X or Y) or the form (not X or not Y) would be interesting? Are there situations in mathematics in which the form of the dilemma argument would be useful? Are there classic examples of mathematical evidence using a dilemma?