Birnbaum considers two experiments that have pairs of respective outcomes—call them “star pairs”—that determine proportional likelihood functions. He then constructs a hypothetical mixture experiment with these two experiments as its components. By the conditionality principle, an outcome of either component experiment has the same as the evidential meaning of the corresponding outcome of the mixture experiment. Now, there is a sufficient statistic that lumps together outcomes of the mixture experiment corresponding to star pair outcomes of the component experiments. By the sufficiency principle, then, these outcomes of the mixture experiment have the same evidential meaning. Given that an outcome of the mixture experiment has the same evidential meaning as a corresponding outcome of a component experiment, and that outcomes of the mixture experiment corresponding to star pair outcomes of the two component experiments have the same evidential meaning as one another, it follows that star pair outcomes of the two component experiments have the same evidential meaning as one another. That is just what the likelihood principle asserts.
Call the two component experiments E and E’, respectively; call the mixture experiment E*; and let (x*, y*) be a "stair pair," with x* an outcome of E and y* an outcome of E’. Then the following diagram depicts the structure of Birnbaum’s proof, using lines to indicate evidential equivalence and denoting above each line which principle is invoked to establish equivalence:
Several objections to this proof have appeared in the statistics literature (e.g. Durbin 1970, Cox and Hinkley 1974, Kalbfleisch 1974, Joshi 1990) and at least one in the philosophy literature (Mayo 2011), but it is still widely accepted. I am suspicious of Birnbaum's proof, but I do not yet have confidence in any precise diagnosis of where it goes wrong.
One objection to Birnbaum's proof toward which I am sympathetic says that the conditionality princple should be understood not as a claim about evidential equivalence, but as a directive about how to analyze experimental results: thus, the conditionality principle says, "analyze experimental results conditional on which experiment was actually performed." Understood in this way, the conditionality principle prohibits the use of the sufficient statistic that lumps together results from experiment E and experiment E', blocking Birnbaum's proof. (Kalbleisch develops a version of this idea in his 1974, but the specific way in which he develops it may be problematic.) Birnbaum denies that the conditionality principle is to be understood as a directive (1962, p. 281 and elsewhere), but it is not clear to me that he has good reasons for doing so.