This is a kind of mixed syllogism in which the major premise is an alternative proposition and the minor premise is a categorical proposition affirming one of the alternatives of the major premise. The conclusion is a categorical proposition that denies the other alternative.
X can’t be both Y and Z.
X is y
Therefore, X is not Z.
Rakesh cannot be born in both January and May.
Rakesh is born in January.
Therefore, Rakesh is not born in May.
This is a valid form of syllogism called Modus Ponendo Tollens. But the other form is invalid which is evident from the following example.
A is not be both P and Q.
A is not P.
Therefore A is Q.
Ashok is not both in Delhi and Lucknow.
Ashok is not in Delhi.
Therefore, Ashok is in Lucknow.
This form of alternative-categorical syllogism is invalid as its premises can both be true but the conclusion be false. It must be noted that alternative categorical syllogism is valid if we affirm one of the alternatives in the minor premise, we may deny the other in the conclusion but not conversely.