The backstab is a guaranteed critical hit from behind for 600% of your maximum health.
The Battalion's Backup, when activated, makes you immune to critical hits and minicrits (you receive regular damage instead) and, on top of that, it provides you with 35% damage resistance.
What happens when you combine the two things? Do you get 26 damage (65% of regular knife damage)? Do you still take damage for 600% health?
Best Answer
Backstabs under the effect of Batallion's Backup still deal a lethal amount of damage, as this video sets out to show:
From http://wiki.teamfortress.com/wiki/Critical_hits:
From: http://wiki.teamfortress.com/wiki/Backstab
(Backstabs deal a base damage of 200% of the target's current health, and triple that [600%] on a crit. If the crit is removed, it will go back down to 200% of the current health; still lethal.)
Battalion's Backup does prevent backstabs from landing a critical hit (600% current health), however it does not reduce the non-crit damage of the backstab which is still lethal (200% current health).
While under the effect of Batallion's Backup, this would simply reduce the damage from 600% to 200% of the player's current health. They could have 1,000,000 health and a non-critical backstab would still deal 2,000,000 damage; a lethal amount. (6,000,000 if it criticals; also lethal)
Whether you deal 2x a target's current health, or 6x a target's current health, they are both lethal amounts. Anything equal to or greater than 1x a target's current health will be lethal.
The Phlogistinator's 75% reduced damage buff does not protect against backstabs.
From: http://wiki.teamfortress.com/wiki/Phlog
Backstabs deal 10% damage to spies using the Dead Ringer. 6x current health will instead be 0.6x current health. 2x current health (a non crit backstab such as one being affected by the Battalion's Backup) will instead deal 0.2x current health.
From: http://wiki.teamfortress.com/wiki/Dead_ringer