Theorem zfauscl 3868

Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3866, we invoke the Axiom of Extensionality (indirectly via vtocl 2602), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
zfauscl.1	|- A e. _V

Assertion

Ref	Expression
zfauscl	|- E.yA.x(x e. y <-> (x e. A /\ ph))

Distinct variable groups:   x,y,A   ph,y

Allowed substitution hint:   ph(x)

Proof of Theorem zfauscl

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfauscl.1	. 2 |- A e. _V
2		eleq2 2098	. . . . . 6 |- (z = A -> (x e. z <-> x e. A))
3	2	anbi1d 438	. . . . 5 |- (z = A -> ((x e. z /\ ph) <-> (x e. A /\ ph)))
4	3	bibi2d 221	. . . 4 |- (z = A -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. A /\ ph))))
5	4	albidv 1702	. . 3 |- (z = A -> (A.x(x e. y <-> (x e. z /\ ph)) <-> A.x(x e. y <-> (x e. A /\ ph))))
6	5	exbidv 1703	. 2 |- (z = A -> (E.yA.x(x e. y <-> (x e. z /\ ph)) <-> E.yA.x(x e. y <-> (x e. A /\ ph))))
7		ax-sep 3866	. 2 |- E.yA.x(x e. y <-> (x e. z /\ ph))
8	1, 6, 7	vtocl 2602	1 |- E.yA.x(x e. y <-> (x e. A /\ ph))

Syntax hints:   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019  ax-sep 3866

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  inex1  3882  bj-d0clsepcl  9384