A lattice is an algebraic structure $(L,\wedge,\vee)$ such that, $\wedge$ and $\vee$ are commutative, associative and abosrbing binary operations, i.e.
$$a \wedge (b\vee a)=a,\quad a\vee(a\wedge b)=a.$$
I want to show $$x\wedge a \wedge b=x \Rightarrow x\wedge b=x.$$
Working from a lattice as a poset, and defining $a\wedge b$ as the essential source of $\{a,b\}$, i.e $$x=a\wedge b \iff (x\leq a) \quad (x\leq b) \quad \forall y\in L (y\leq a \text{ and } y\leq b)\Rightarrow y\leq x.$$ Then the result follows. But how do I show this without a relation
If $x\wedge a \wedge b = x$, then $$\begin{align*} x\wedge b &= (x\wedge a\wedge b) \wedge b &\text{(substitution)}\\ &= (x\wedge a)\wedge (b\wedge b)&\text{(associativity)}\\ &= (x\wedge a) \wedge b&\text{(idempotency)}\\ &= x\wedge a \wedge b & \text{(associativity)}\\ &= x. &\text{(substitution)} \end{align*} $$