It's technically true, if your definition of a #furry is someone who bases a significant portion of their #identity on a fan of #anthropomorphic #creatures (the "significant portion of their identity" phrasing is to nip any "#everyone is at least #somewhat furry" arguments in the bud). Under that definition, #bronies are absolutely a #subset of furries. However, there are enough cultural differences that treating #MLP fans as just a #unique partition of #furries who happen only to care about #ponies just isn't accurate.

(Demographic note: this is all from an uncaged pocket #monster who is really only an active part of the proudly bad side of the pony fandom and associates with lots of furries online but has no desire to go to a fur #con or make a real 'sona[0])

First, the #pony fandom is much younger than the furry #fandom.
There are probably some OG furries around, but the vast majority of furries (especially around here) are younger than I and know that the fandom has existed long before they were likely born. It probably was around before the furries I know IRL (who are a couple years older than I) were born.
The #horse fandom has only existed in its current form since #2010, which means I could have been one of the "since the very start" members if I had different drinking buddies a year and a half before I watched my first pony episodes, or even a "founding" member if I browsed #chans instead of mastering the art of fire-breath.

About half of the cultural divide between furries and bronies is simply due to how new and seemingly temporary the pony fandom is and the fact that the MLP fandom exploded in popularity and has been gradually shrinking ever since (with the infall of new bronies being much smaller than the outflow of burned-out bronies). There may have been times when the furry community grew as fast as MLP in 2011 (I don't know, but there may have been some relevant #Disney movies—#LionKing), but there was an existing core community before the new fans left and a modified core existed after the fair-weather fans left. MLP had no such store of cultural knowledge, and (especially with the G5 rumors) has the constant threat that the fandom will probably dissolve once #Hasbro or #DHX stops making as much #money as they'd like.

Ponies' reliance on a single defined #IP also creates the second difference: there is no expectation that you make a pony OC (in fact, it's sometimes mocked), while the furry community is all about your 'sonas. It's the difference between living in Hasbro's world of #Equestria (or forking it when they cancel the show or make intolerable changes) and making your own characters to interact in the real world as the starting point. See note [0].

Going back to the youth of the pony fandom, the broken delusion that there ever was a unified fandom still leaves its tracks in how sub-communities interact and have control. More importantly, I've seen the following difference in attitude from "normal" folks in regard to the two fandoms: furries are "weird, please don't do it in my face, but w/e"; bronies mostly "who doesn't like MLP? they just take it too far". It seems to be how the two groups interact that the attitude difference is apparent (though it's calmed down compared to 2014): bronies either have dual fandom citizenship (in both directions) with the fur community, but furries are either neutral or hostile to bronies.
I suspect this has to do with the "respectable" subset of furries generally winning the major PR and the "bad" side mostly accepting that fact, so they can coexist. There still is an active #respectability politics push in the brony fandom (that may not resolve by the time it falls apart), which fuels the bad side to proudly proclaim they are #horsefuckers just to piss off those who want to improve the image of the fandom. I have seen orders of magnitude more #jokes about #bestiality aimed at bronies than furries (though this was very different in 2005, if my understanding is accurate), and I suspect all the "proud #horsefucker" #shitposts are a major reason why. The majority of the bad side of furries keep their shitposts and humor on the DL and are content to let the respectable half take the spotlight, while the "#proud pony #degenerates" love to remind the world that the family-friendly half of the fandom is just good PR cover for the trolls as well.

I think this leads me to my conclusion & tl;dr:
The chan origins and massive #growth of the brony fandom imprinted a very different set of initial conditions on the pony fandom from the evolution of furries, so the fandoms are not #miscible with one another, even though they share many members; the growth & presumed ephemerality of MLP itself render futile any attempt to resolve any major fandom splits (either a clean split or agreeing to merge).

A more accurate statement is that "bronies are furries with a single canonical IP to reference", which is too major a difference for an "all #Dalmatians are dogs, but not all #dogs are Dalmatians" #analogy.
#Java analogy: Bronies and Furries implement the same interface, but bronies do not inherit from the furry class, even though they have many of the same methods and attributes.

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[0] If anything, @USBloveDog (formerly @USBhump) was a novelty RP that ditched the RP and became the avatar for my id, but it's not "the real me" in the way many furries view their sona or bronies would view their OC. Semi-permeable RP is how I like to run online accounts.
[1] Hopefully this was nice and coherent and I didn't miss too many points.
#essay