let H be ZF-formula; :: thesis: for x, y being Variable holds
( H is being_membership iff H / x,y is being_membership )
let x, y be Variable; :: thesis: ( H is being_membership iff H / x,y is being_membership )
thus ( H is being_membership implies H / x,y is being_membership ) :: thesis: ( H / x,y is being_membership implies H is being_membership )

proof

given x1, x2 being Variable such that A1: H = x1 'in' x2 ; :: according to ZF_LANG:def 11 :: thesis: H / x,y is being_membership
ex z1, z2 being Variable st H / x,y = z1 'in' z2 by A1, Th169;
hence H / x,y is being_membership by ZF_LANG:16; :: thesis: verum

end;

assume A2: H / x,y is being_membership ; :: thesis: H is being_membership
( 1 <= 3 & 3 <= len H ) by ZF_LANG:29;
then ( 1 <= 1 & 1 <= len H & dom H = Seg (len H) ) by FINSEQ_1:def 3, XXREAL_0:2;
then A3: ( (H / x,y) . 1 = 1 & y <> 1 & 1 in dom H ) by A2, Th148, FINSEQ_3:27, ZF_LANG:36;
then H . 1 <> x by Def4;
then 1 = H . 1 by A3, Def4;
hence H is being_membership by ZF_LANG:42; :: thesis: verum