In conjunction with a simple rule of disjoint reference (which stipulated that any pronoun following a noun phrase (NP) antecedent in the same sentence has disjoint reference with it, the rule applying anywhere unless it is blocked), co-reference is acceptable in the following sentences, because the SSC blocks application of this disjoint reference rule: The TSC (which essentially blocks transformational and binding rules from applying across clause boundaries) would not block disjoint reference in (1) and (2), hence the need for the SSC.
Replacing the pronouns in (1) and (2) with reciprocals shows how the SSC blocks the application of each movement, hence the impossibility of the reciprocals referring back to "The footballers" in (3) and (4): Notice that when the DP-internal subject is removed, each movement is not blocked from applying: An empirical problem for the SSC is the failure of disjoint reference to apply in a sentence like (6), where there is no specified subject blocking its application: The SSC also made correct predictions for certain binding data with respect to control verbs.
Similarly, in (9), PRO is a specified subject for "we", thus blocking disjoint reference, so that "we" can corefer with "us" in the non-finite clause.
Similar examples hold for subject control verbs like "persuade": *Theyi promised Billj [PROi to kill themi] vs Billj promised themi [PROj to kill themi], and subject raising verbs like "seem": *Theyi seem to Billj [ti to like themi] (where the trace is not specified with respect to "we" thus disjoint reference applies) vs Wei seem to Billj [ti to like himj] (where the trace is specified with respect to "Bill" so that disjoint reference is blocked).
The subsequent binding conditions A and B of Chomsky (1981) essentially replaced the SSC (along with the TSC), and it is no longer a part of the toolkit of current researchers.