Birnbaum proved in 1962 that (S) and (C) entail (L), and in 1964 that (M) and (C) entail (L), where (M) is strictly weaker than (S). Evans et al. proved in 1986 that (C) alone entails (L). In this post I’ll explain the Evans et al. proof in a rough, qualitative way.
The clearest formulation of (C) for present purposes is the one Birnbaum provides in his (1972):
Conditionality (C): If h(x) is an ancillary statistic, then Ev(E,x)=Ev(Eh,x), where h=h(x).
(See the previous post for an explication of the term “ancillary statistic.”)
The Evans et al. proof starts with an arbitrary experimental outcome (E1,x*). It then constructs a hypothetical outcome (B,h) that has the same likelihood function. The next step is to construct a cross-embedded experiment that I will call CE(E1,B). This cross-embedded experiment has two maximal ancillary statistics: the variable of which x* is an instance is ancillary with respect to the variable of which h is an instance, and vice versa. Thus, it is possible to apply (C) to CE(E1,B) twice to establish that (CE(E1,B),(x*,h)) is evidentially equivalent to both (E1,x*) and (B,h), and thus that (E1,x*) and (B,h) are evidentially equivalent to one another. The next step is to apply the same trick to an arbitrary experimental outcome (E2,y*) with the same likelihood function as (E1,x*), using the same hypothetical (B,h). From the fact that (E1,x*) and (E2,y*) are both evidentially equivalent to (B,h), it follows that they are evidentially equivalent to one another. Because the only constraint placed on (E1,x*) and (E2,y*) in this construction is that they must have the same likelihood function, the likelihood principle follows immediately.
Allowing text boxes to represent experimental outcomes and lines to represent evidential equivalence (established by (C)), this proof can be represented by the following diagram: