The abs and core is the component that connects your whole kinetic chain. Building a strong core will lead to you becoming stronger in all exercises.
Whether it be a static isometric hold or a dynamic movement, your core is always being activated.
A bridge is only as strong as its weakest link, and your body works in a similar fashion. Many times your core is the weak link.
When your perform the push-up , your body is essentially performing a plank the entire time.