In terms of Dougall's $_2H_2$ series identity and the series rearrangement method, we establish an interesting symmetric formula for hypergeometric series. Then it is utilized to derive a known nonterminating form of Saalschütz's theorem. Similarly, we also show that Bailey's $_6ψ_6$ series identity implies the nonterminating form of Jackson's $_8φ_7$ summation formula. Considering the reversibility of the proofs, it is routine to show that Dougall's $_2H_2$ series identity is equivalent to a known nonterminating form of Saalschütz's theorem and Bailey's $_6ψ_6$ series identity is equivalent to the nonterminating form of Jackson's $_8φ_7$ summation formula.