Theorem zfauscl 4135

Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4133, we invoke the Axiom of Extensionality (indirectly via vtocl 2803), 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 2251	. . . . . 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 1834	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
6	5	exbidv 1835	. 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 4133	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8	1, 6, 7	vtocl 2803	1 |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1361   = wceq 1363   E.wex 1502   e. wcel 2158   _Vcvv 2749

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169  ax-sep 4133

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751

This theorem is referenced by:  inex1  4149  bj-d0clsepcl  14973