  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
Saalsch\"{u}tz's theorem. Similarly, we also show that Bailey's $_6\psi_6$
series identity implies the nonterminating form of Jackson's $_8\phi_7$
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 Saalsch\"{u}tz's theorem and Bailey's $_6\psi_6$ series
identity is equivalent to the nonterminating form of Jackson's $_8\phi_7$
summation formula.
