Theorem zfauscl 4084

Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4082, we invoke the Axiom of Extensionality (indirectly via vtocl 2766), 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 2221	. . . . . 6 |- ( z = A -> ( x e. z <-> x e. A ) )
3	2	anbi1d 461	. . . . 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 1804	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
6	5	exbidv 1805	. 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 4082	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8	1, 6, 7	vtocl 2766	1 |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1333   = wceq 1335   E.wex 1472   e. wcel 2128   _Vcvv 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139  ax-sep 4082

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714

This theorem is referenced by:  inex1  4098  bj-d0clsepcl  13460