Then we need the use of two laws :
1) Universal Elimination ,and then
2) Modus Ponens.
in the following way:
for all ,x ( if x is a field ,then x is a ring) [this is a fact in abstract alebra]
Now by using the law of Universal Elimination ,by puting x=A, we have:
If A is a field ,then A is a ring .
Then we have to add
but, A is a field , to conlcude
A is a ring
The same thing we can say for B
But certainly :
If A is a field, A is also a ring.
B is a field, hence B is a ring.
Tell us nothing.
But to come back to our problem ,apart from the law of transitivity ,what other theorems or axioms are involved in the proof??
Also Shadow mentioned the use of M.Tolens .Where is that law involved in the proof??