Theorem zfauscl 3965

Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3963, we invoke the Axiom of Extensionality (indirectly via vtocl 2674), 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. y A. 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 2152	. . . . . 6 |- ( z = A -> ( x e. z <-> x e. A ) )
3	2	anbi1d 454	. . . . 5 |- ( z = A -> ( ( x e. z /\ ph ) <-> ( x e. A /\ ph ) ) )
4	3	bibi2d 231	. . . 4 |- ( z = A -> ( ( x e. y <-> ( x e. z /\ ph ) ) <-> ( x e. y <-> ( x e. A /\ ph ) ) ) )
5	4	albidv 1753	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
6	5	exbidv 1754	. 2 |- ( z = A -> ( E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
7		ax-sep 3963	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8	1, 6, 7	vtocl 2674	1 |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1288   = wceq 1290   E.wex 1427   e. wcel 1439   _Vcvv 2620

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-ext 2071  ax-sep 3963

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-v 2622

This theorem is referenced by:  inex1  3979  bj-d0clsepcl  12093