Just to add a point to what gas path has said:
The 777 fly-by-wire system requires triple channel primary flight computers and servos to be engaged at all times (regardless of autopilot engagement). Prior to autopilot engagement the PFCs respond to signals from the yoke/pedals. This effectively means that the 777 is flying with triple channel autopilot computers and servos engaged even before the autopilot engage button is pressed. When the pilot engages the autopilot the yoke/pedal signals are replaced by the autopilot command signals. Since triple channel PFCs and servos are already engaged triple channel command signals are required to drive/monitor them. So one button is used to engage all three autopilots command channels simultaneously.
This is fundamentally different to the way typical non fly-by-wire aircraft like the 747 behave. In normal operation only a single autopilot is engaged. Triple channel is only used when increased redundancy is required for automatic approach/landing.