Egalitarianism is about justice. Justice in the sense that most people--but not the very rich--understand: fairness, decency, reasonableness. Justice in the sense of the Golden Rule.
In our current society based on class inequality, the rich ruling upper class perverts the word "justice" to mean something grotesque that fits smoothly with the fundamental injustice of there being a few enormously wealthy and powerful haves ruling over and dominating the vast majority of have-nots. The rich call anything unjust if it prevents the rich haves from living in luxury at the expense of the have-nots having to make do with very little. We say its unjust to allow the rich haves to do this.
Egalitarianism says it's justice to base the economy on the principle of "From each according to reasonable ability, to each according to need or reasonable desire with scarce things rationed equitably according to need." The rich say that's an unjust principle, that it is "theft" to prevent a billionaire from owning billions of dollars worth of socially produced wealth.
Egalitarianism is about justice based on genuine democracy. This means that ordinary people--the people who value equality and mutual aid and who take the Golden Rule seriously, i.e. egalitarians--decide what's just and what's not just. All the egalitarians in a community, and only they, have the right to participate as equals in the local assembly, which is the highest authority for everybody in its community. What good and decent people in their Local Assembly decide is fair and reasonable and in keeping with the Golden Rule is the law of the land for that community. This is real justice. What we have now from the so-called "justice" branch of government is the injustice that the rich call "justice."
Virtually everybody today has experienced the fake "justice" of the rich. It was described very well in 1894 by Anatole France:
"In its majestic equality, the law forbids rich and poor alike to sleep under bridges, beg in the streets and steal loaves of bread."
Today we might add that in its same "majestic equality," the law calls for rich and poor alike to be evicted from their homes when they can't pay the rent, and for rich and poor alike to be made homeless when they can't meet a mortgage payment. Both rich and poor alike will be arrested for illegal drug possession if they make their purchase on the street instead of in a five-star hotel suite. It is illegal for both rich and poor alike to smoke tobacco in their federally funded low-income housing project home.
Both rich and poor alike, in the "majestic equality" of today's "justice," will be severely punished for desertion if they desert from their post in the U.S. military after 'voluntarily' enlisting in it to get a job (the "poverty draft") or to get citizenship. And speaking of jobs, both rich and poor alike have to meet the same requirements to get hired for the same job, be it a minimum wage job or one requiring a Ph.D. or even one of those "it's not what you know but who you know" jobs. And it makes no difference if you're rich or poor, you will be arrested for trespassing if you refuse to leave the premises after being fired from your job for organizing workers to demand better pay and working conditions. And the police arrest rich and poor people alike whenever they try to physically block a scab from crossing their picket line during a strike.
The "majestically impartial" law today doesn't care if you own the hospital or are an orderly in it: either way, if you are sick and need expensive medical care and you don't have a good enough health insurance policy or you can't pay the medical bill directly, then you don't get the health care you need.
And no matter if you're rich or poor, if you're charged with a capital crime in a jurisdiction that has the death penalty for that crime, not having an expensive high-powered attorney will greatly increase the probability that you will be executed even if you're innocent, as most likely happened to Ledell Lee who was executed on 20 April 2017, and whose attorney, Craig Lambert, later admitted in an affidavit obtained since Lee’s execution, signed by Lambert in October: “I was struggling with substance abuse and addiction in those years. I attended inpatient rehab. Ledell’s case was massive and I wasn’t in the best place personally to do what was necessary.”
The "justice" of the rich always results in the rich staying rich and powerful and comfortable, while the poor remain poor and powerless and uncomfortable. The judges have huge law libraries whose books contrive to make this phony "justice" seem just. But it's all a sham. Anybody who has been subjected to this phony "justice" knows it's not just, even if they have a hard time saying exactly why.
In an egalitarian society, what ordinary people know is fair and decent and just is what determines the laws and policies. Real justice can finally prevail.