Relationship between incomplete beta and hypergeometric 2F1

Theorem: The following formula holds: $$B_x(a,b)=\dfrac{x^a}{a} {}_2F_1(a,1-b;a+1;x),$$ where $B_x$ denotes the incomplete beta function and ${}_2F_1$ denotes the hypergeometric pFq.

Proof: █