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

proof

given H1 being ZF-formula such that A1: H = 'not' H1 ; :: according to ZF_LANG:def 12 :: thesis: H / x,y is negative
H / x,y = 'not' (H1 / x,y) by A1, Th170;
hence H / x,y is negative by ZF_LANG:16; :: thesis: verum

end;

assume A2: H / x,y is negative ; :: thesis: H is negative
( 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 = 2 & y <> 2 & 1 in dom H ) by A2, Th148, FINSEQ_3:27, ZF_LANG:37;
then H . 1 <> x by Def4;
then 2 = H . 1 by A3, Def4;
hence H is negative by ZF_LANG:43; :: thesis: verum