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This paper presents a partially synchronous BFT consensus protocol powered by BBCA, a lightly modified Byzantine Consistent Broadcast (CBC) primitive. BBCA provides a Complete-Adopt semantic through an added probing interface to allow either aborting the broadcast by correct nodes or exclusively, adopting the message consistently in case of a potential delivery. It does not introduce any extra type of messages or communication cost to CBC. BBCA is harnessed into BBCA-CHAIN to make direct commits on a chained backbone of a causally ordered graph of blocks, without any additional voting blocks or artificial layering. With the help of Complete-Adopt, the additional knowledge gained from the underlying CBC completely removes the voting latency in popular DAG-based protocols. At the same time, causal ordering allows nodes to propose blocks in parallel and achieve high throughput. BBCA-CHAIN thus closes up the gap between protocols built by consistent broadcasts (e.g., Bullshark) to those without such an abstraction (e.g., PBFT/HotStuff), emphasizing their shared fundamental principles. Using a Bracha-style CBC as an example, we fully specify BBCA-CHAIN with simplicity, serving as a solid basis for high-performance replication systems (and blockchains).