Theorem zfauscl 4164

Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4162, we invoke the Axiom of Extensionality (indirectly via vtocl 2827), 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 2269	. . . . . 6 |- ( z = A -> ( x e. z <-> x e. A ) )
3	2	anbi1d 465	. . . . 5 |- ( z = A -> ( ( x e. z /\ ph ) <-> ( x e. A /\ ph ) ) )
4	3	bibi2d 232	. . . 4 |- ( z = A -> ( ( x e. y <-> ( x e. z /\ ph ) ) <-> ( x e. y <-> ( x e. A /\ ph ) ) ) )
5	4	albidv 1847	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
6	5	exbidv 1848	. 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 4162	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8	1, 6, 7	vtocl 2827	1 |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176   _Vcvv 2772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187  ax-sep 4162

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  inex1  4178  bj-d0clsepcl  15865