Valid Syllogism in Figure I needs Affirmative Minor Premise and Universal Major Premise

Theorem
Let $Q$ be a valid categorical syllogism in Figure I.

Then it is a necessary condition that:
 * The major premise of $Q$ be a universal categorical statement

and
 * The minor premise of $Q$ be an affirmative categorical statement.

Proof
Consider Figure I:

Let the major premise of $Q$ be denoted $\text{Maj}$.

Let the minor premise of $Q$ be denoted $\text{Min}$.

Let the conclusion of $Q$ be denoted $\text{C}$.

$M$ is:
 * the subject of $\text{Maj}$
 * the predicate of $\text{Min}$.

So, in order for $M$ to be distributed, either:
 * $(1): \quad$ From Universal Categorical Statement Distributes its Subject: $\text{Maj}$ must be universal

or:
 * $(2): \quad$ From Negative Categorical Statement Distributes its Predicate $\text{Min}$ must be negative.

Suppose $\text{Min}$ is a negative categorical statement.

Then by Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative:
 * $\text{C}$ is a negative categorical statement.

From $(2)$:
 * $P$ is distributed in $C$.

From Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise:
 * $P$ is distributed in $\text{Maj}$.

From Negative Categorical Statement Distributes its Predicate:
 * $\text{Maj}$ is a negative categorical statement.

Thus both:
 * $\text{Min}$ is a negative categorical statement
 * $\text{Maj}$ is a negative categorical statement.

But from No Valid Categorical Syllogism contains two Negative Premises, this means $Q$ is invalid.

Thus $\text{Min}$ is not a negative categorical statement in Figure I.

As $\text{Min}$ needs to be an affirmative categorical statement, $M$ is not distributed in $\text{Min}$.

From Middle Term of Valid Categorical Syllogism is Distributed at least Once, this means $M$ must be distributed in $\text{Maj}$.

As $M$ is the subject of $\text{Maj}$ in Figure I, it follows from $(1)$ that:
 * $\text{Maj}$ is a universal categorical statement.

Hence, in order for $Q$ to be valid:
 * $\text{Maj}$ must be a universal categorical statement
 * $\text{Min}$ must be an affirmative categorical statement.