Interacting selectively and non-covalently with the BH4 domain of a protein of the Bcl-2 family. All anti-apoptotic proteins contain BH1 and BH2 domains; some also contain an additional N-terminal BH4 domain, which is almost never seen in pro-apoptotic proteins. Loss of the BH4 domain can diminish or abrogate anti-apoptotic function or even impart outright death-promoting properties to the protein.