The OP is using $B_{RF}$ to refer to the passband of the front-end filter ahead of the mixer, and $B_c$ to refer to actual occupied bandwidth of the signal. With that definition, the OP is correct with his formula, as long as $B_{RF}$ is given by the rejection requirement for the alias region and not a passband for the signal of interest: for example a -50 dB bandwidth rather than the typical use of a -1 dB or -3 dB bandwidth. Further such filters can and often are asymmetrical favoring rejection where it is needed, so this approach is not the best way to define the RF filter requirement. Instead I recommend defining the passband to pass the channel of interest with a maximum ripple and group delay variation requirement, and a stop band defined with sufficient rejection as needed to reject the alias region.
I suspect that the text intended for $B_{RF}$ to represent the maximum possible theoretical bandwidth of the modulated RF signal itself (with use of impossible to realize brick-wall filters to reject the alias region). When $B_{RF}> 2f_{IF}$ the spectrums will overlap, meaning the spectrum of our RF signal of interest, with the spectrum of whatever is in the alias region, as both will land in the IF frequency region using a real down-converter. Even if there is no signal in the alias region, if the noise floor there is the same as the noise in our bandwidth of interest, the total noise will sum in power and a 3 dB penalty in noise figure will result.
Filtering out the local oscillator (LO) feedthrough as Tim Wescott points out in his answer is certainly a benefit that we would also achieve with this condition, as we don't want the LO to reach the antenna and radiate. This is a feature but not a theoretical hard requirement as the text, based on the theoretical limits, not practical application, is presenting. A case in point is a Zero-IF implementation where such filtering of the LO is not possible (the LO is centered on the RF bandwidth), in which case other techniques are used such as the reverse isolation of the front-end low noise amplifier, in addition to carrier feedthrough nulling techniques at the mixer, and the additional use of isolators if further reverse isolation is needed. These same techniques can be applied here in an super-heterodyne architecture with an IF frequency, whereas the deleterious effects of aliasing cannot be avoided in a single mixer architecture if $B_{RF}>2f_{IF}$.
For the purposes of the anti-alias filter design, it is generally better to define a bandwidth region that you would pass, a bandwidth region that you would reject, and what is in between is the transition band. The complexity of the filter is driven by the transition band that we can allow between these two.
That said, the OP's interpretation of the frequency locations and bandwidths for rejection are correct as drawn.
As far as filter design requiements, The passband shoud be defined to pass $B_{RF}$ with whatever distortion requirements are necessary, and the stop band is to start at $2f_{IF} - B_{RF}/2$ offset from $f_c$, and should be at least $B_{RF}$ wide, and everything else is "don't care", as I show this in the graphic below.
With that understood, and bringing in practical considerations, we see that the real requirement is $B_{RF}>2KB_{IF}$ with $K>1$ as a complexity constant. The closer $K$ is to 1, the more impossible it is to design an actual filter since there will be no transition band. For example below shows the filter requirement with $K=1.1$. Cases like this with small $K$ is a motivation for using an image reject mixer.
