Axioms for Expected Utility:
Let $\succ $ be a binary relation on $X$.
A1. $\succ $ is asymmetric and negatively transitive.
A2. Independence of Irrelevant Alternatives: If $p,q,r \in X$ and if $\alpha $ is a number such that $1\geq \alpha > 0$, then $p\succ q \Leftrightarrow \alpha p + (1-\alpha )r \succ \alpha q +(1-\alpha )r.$
A3. Archimedean Axiom: If $p,q,r \in X$ are such that $p\succ q\succ r$, then there are numbers $\alpha$ and $\beta$ satisfying $1> \alpha > \beta > 0$ such that $\alpha p+(1-\alpha )r\succ q\succ \beta p+(1-\beta )r.$
Implications of the Axioms
The proof of the expected utility theorem makes use of a different set of behavioural rules. If $\succ$ on $X$ satisfies $A1,A2,A3$, then $L1,L2,L3$ all hold.
L1. Let $\alpha$,$\beta \in \mathbb{R}$ be any numbers satisfying $1\geqslant \alpha > \beta \geq 0$. Then for any $p$,$q \in X$, $p\succ q\Leftrightarrow \alpha p+(1-\alpha)q\succ \beta p+(1-\beta)q$.
L2. If $p,q,r \in X$ are such that $p \succ r$ and $p\succeq q\succeq r$, then there is a unique number $\alpha^*$ such that $q\sim \alpha ^*p + (1-\alpha^* )r$.
L3. If $p,q\in X$ are such that $p \sim q$ then $\alpha p+(1-\alpha )r\sim \alpha q+(1-\alpha)r$ for any $r\in X$ and any $0\leq \alpha \leq 1$.
How do I use the axioms to show that they lead to L1,L2,L3?
The proofs are rather lengthy and messy, you can take a look at the proof of Theorem 8.2. in the book Utility theory for decision making by Peter C. Fishburn, which essentially proves the expected utility theorem from A1-A3 and proves L1-L3 along the way.