let E be non empty set ; :: thesis: for x being Variable
for H being ZF-formula
for f being Function of VAR ,E holds
( ( f in St H,E & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E ) ) iff f in St (All x,H),E )
let x be Variable; :: thesis: for H being ZF-formula
for f being Function of VAR ,E holds
( ( f in St H,E & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E ) ) iff f in St (All x,H),E )
let H be ZF-formula; :: thesis: for f being Function of VAR ,E holds
( ( f in St H,E & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E ) ) iff f in St (All x,H),E )
let f be Function of VAR ,E; :: thesis: ( ( f in St H,E & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E ) ) iff f in St (All x,H),E )
A1: All x,H is universal by ZF_LANG:16;
then A2: St (All x,H),E = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR ,E st X = St (the_scope_of (All x,H)),E & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
bound_in (All x,H) = y ) holds
g in X ) ) } by Lm3;
All x,H = All (bound_in (All x,H)),(the_scope_of (All x,H)) by A1, ZF_LANG:62;
then A3: ( x = bound_in (All x,H) & H = the_scope_of (All x,H) ) by ZF_LANG:12;
thus ( f in St H,E & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E ) implies f in St (All x,H),E ) :: thesis: ( f in St (All x,H),E implies ( f in St H,E & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E ) ) )

proof

assume A4: ( f in St H,E & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E ) ) ; :: thesis: f in St (All x,H),E
reconsider v = f as Element of VAL E by Def2;
for X being set
for h being Function of VAR ,E st X = St (the_scope_of (All x,H)),E & h = v holds
( h in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> h . y holds
bound_in (All x,H) = y ) holds
g in X ) ) by A3, A4;
hence f in St (All x,H),E by A2; :: thesis: verum

end;

assume f in St (All x,H),E ; :: thesis: ( f in St H,E & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E ) )
then A5: ex v5 being Element of VAL E st
( f = v5 & ( for X being set
for f being Function of VAR ,E st X = St (the_scope_of (All x,H)),E & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
bound_in (All x,H) = y ) holds
g in X ) ) ) ) by A2;
hence f in St H,E by A3; :: thesis: for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St H,E
let g be Function of VAR ,E; :: thesis: ( ( for y being Variable st g . y <> f . y holds
x = y ) implies g in St H,E )
assume for y being Variable st g . y <> f . y holds
x = y ; :: thesis: g in St H,E
hence g in St H,E by A3, A5; :: thesis: verum