How do I prove symmetry without a defined set?

Question

I have a formula:

∀x,y, z(xRy ∧ xRz → yRz)

If the formula holds for a relation, then the relation is Euclidean.

If a relation is Euclidean and reflexive, what are the steps for proving it is also symmetric and transitive?

I've worked a little with relations but usually I've worked with defined sets and have been provided parameters such as:

R1 = {(x,y) | x + y > 5} ⊆ R × R

However, the only information provided for this is the formula and nothing else.

I'm not sure how to approach this and would love to understand. I'm sure I'm missing something simple here and would really appreciate a few worked examples of similar problems, so I can work this one out myself.

Answer 1

1. Symmetry. Let $z=x.$ Then $xRy\to (xRy\land xRx)\to (xRy\land xRz)\to yRz\to yRx.$

2. $(xRy\land yRz)\to (yRx\land yRz)$...(by 1.Symmetry)...$\to xRz$... (by Euclidean on $y,x,z$ instead of $x,y,z$).