Trivial answer: no, because you cannot center antimagic field on a pebble
Antimagic field can only be emanated from the caster him- or herself. You cannot cast it on other creatures or objects, the way you can with, e.g., light. Thus, the sequence you describe cannot take place.
Arcane archers1 and master abjurers2 can stick antimagic field onto other creatures, but not onto objects.
Trivial answer: no, the prismatic field destroys anything you try to bring in
Still, if you were an arcane archer or master abjurer, you could presumably tag some tiny-and-harmless critter with antimagic field (the master abjurer is much nicer about this process) and carry that instead of your pebble.
The violet veil is still going to just destroy it. It won’t make it to you, and the antimagic field will be annihilated along with the kitten (and you become a terrible person, if you were a master abjurer and not already).
Trivial answer: no, the antimagic field is in the prismatic sphere with you
OK, fine: assume an indestructable kitten, or at least one immune to the various effects of prismatic sphere.3
So you have your antimagic field-emanating kitten, you step out of the field (or never entered it, if you were a heartless bast— I mean, arcane archer), and you cast prismatic sphere. You now step towards the kitten, and the antimagic field is held at bay by your prismatic sphere. You step close enough to pick up the kitten,4 at which point your prismatic sphere is wholly enclosed inside an antimagic field. What happens?
The kitten is now inside your prismatic sphere. The prismatic sphere’s edge is a barrier through which the antimagic field cannot penetrate, but our amazing kitten can. Since the source of the antimagic field is now inside the prismatic sphere, it affects you, suppressing your magic, including the prismatic sphere, and freeing the antimagic field to extend to its full radius (rather than being limited by the inside of the prismatic sphere). Your prismatic sphere will return if and only if you step away from the kitten, allowing it to stay outside your prismatic sphere.
Real answer: antimagic field does not block magic, only suppresses it
Ultimately, the thing you were trying to do here is possible. The archmage’s Mastery of Shaping, for instance, could leave a 5-foot hole in the middle of the emanation that is centered on you, leaving you magical, with antimagic field around you. Unfortunately, that’s largely worthless, since magic can pass through an antimagic field. It doesn’t stop magic, just suppresses it while it’s in the antimagic field.
In game terms, antimagic field does not block the line of effect needed to target spells. So you are just as vulnerable to spells cast at you as you were without the antimagic field. Someone standing outside your shell can cast fireball at you, and the little red bead will disappear when it hits the shell – and reappear on your side of the shell, before hitting you in the face.5 Then it blows up, filling your hollow with fire, leaving your antimagic field-filled shell free of any flames, and yet more flames outside the shell.
All it really does is suppress the magic of those standing very close to you – which isn’t awful by any means, but it is a far cry from the invulnerability to spells you might imagine it is.
Also, casting prismatic sphere at this point does an interesting – but not particularly useful – thing. It limits the size of your antimagic field, basically defeating the Widen Spell metamagic you attempted to put on it. Instead of a 20-ft.-radius antimagic field with a 5-ft.-square hole in the middle, you end up with a 10-ft.-radius antimagic field with a 5-ft.-square hole in the middle, and a prismatic sphere around that. Even if you shrunk the prismatic sphere somehow, so it was inside the 5-ft.-square hole, your antimagic field won’t happen outside it.
If you do get a prismatic sphere with an antimagic field outside it, which can be done in more convoluted ways than I feel like getting into, spells can still go right through the antimagic field to attempt to cancel your prismatic sphere. But the prismatic sphere would not “wink out” because of the antimagic field around it, because antimagic field does not block magic, just suppresses it in the specific area covered by the field.6
Arcane archers can shoot an arrow that sticks a spell that ordinarily can only emanate from you, onto a target creature, who emanates it instead of you. The class feature was almost certainly written specifically for antimagic field; there are very few other spells where this would be desirable, but for antimagic field it is very desirable.
A “master abjurer” here is an abjurer who takes the master specialist prestige class in Complete Mage – their major school esoterica class feature allows them to cast abjurations that are normally centered on you (e.g. antimagic field) as a touch-attack spell that emanates from the touched creature. Again, antimagic field was almost-certainly the spell they had in mind when they wrote this.
Arcane archer, that doesn’t make what you did any kind of okay.
For the purposes of this answer, I am ignoring the fact that antimagic field and prismatic sphere have exactly the same size, and assuming the antimagic field is larger; assume Widen Spell was applied to antimagic field if it makes you feel better.
And if you are an arcane archer, you deserve it.
Even if antimagic field did block magic, it’s far from clear to me that the ambient magical energy necessary for spells wouldn’t still exist in the hollow – after all, I can have air inside a container whose walls block air. This is a setting detail that is not addressed by the general rules, and you would have to ask your DM.
Spells which keep them/allies out of range, if the enemy is dangerous in melee
Mobility via teleportation (Misty Step, Dimension Door, Teleport) or increased speed (Haste, Fly, Investiture of Wind) can keep them out of reach of the enemy, if it's strong in melee.
Spells which protect them/allies from ranged attacks, if the enemy is dangerous at range
Some spells can make them or their allies harder to hit with ranged weapon attacks (Warding Wind, Investiture of Wind, Haste).
Others can provide total cover against ranged attacks (Wall of Stone, Wall of Force).
Spells which heal themself/allies fighting the enemy
Healing spells or spells which remove/prevent debilitating conditions will keep their martial allies in the fight. (e.g. Heal, Greater Restoration, Dispel Evil and Good)
Spells which provide powerful nonmagical attacks at range
Shapechange or True Polymorph can allow the spellcaster to transform themself into a creature with strong non-magical ranged attacks, such as a Giant (rocks) or a Dragon (breath).
Situationally, Control Water could allow you to try to drown the enemy by causing water to flow outside the area of the spell.
Necromancy spells (watch out for uncontrolled zombies)
Undead that the spellcaster has raised with spells such as Animate Dead or Create Undead will not become inanimate in the Antimagic Field - but the spellcaster's control over them will lapse. Given that the undead will at that point be right next to the enemy, that's probably not all that bad.
Trap them in a Prismatic Wall
Prismatic Wall is explicitly immune to Antimagic Field, and can be shaped to contain such an enemy. Since they've got an Antimagic Field up, they must drop it before they can cast any teleportation spells to escape this trap.
Wish the Antimagic away
The only spell which is capable of directly dispelling the Antimagic Field is Wish. Speaking as a DM, "Beat this specialised level 8 spell" is probably not the level of power at which I would start to bring in unintended consequences of the Wish. This does risk a 33% chance to never be able to cast Wish again, so it is a steep cost.