That's the Law of Syllogism
Let's go through the list.
Substitution property
This is a very general rule in math that allows us to substitute things that are equal for one another.
Symmetric property
Abstractly, a relation is a set of ordered pairs. A relation R is symmetric when
[tex](a,b) \in R \iff (b,a) \in R[/tex]
Usually we talk about equality or congruence being symmetric: a=b implies b=a.
Law of Syllogism
That's a rule of logic that says if a implies b and b implies c then a implies c. Again it's a very general rule that applies across all mathematics.
Law of Detachment
This one is more commonly known as modus ponens. It's a general rule of formal logic, applying across all mathematics.
If we know two propositions p implies q and p then we can conclude a third proposition, q.
Formal logic acts at a purely syntactic level. It is a mechanistic pushing around of symbols on the page. Nonetheless, in the real world when the premises of an argument are true, the conclusions drawn as a result of formal logic will be true as well.
Answer: Law of Syllogism
