Theorem zfauscl 4783

Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4781, we invoke the Axiom of Extensionality (indirectly via vtocl 3259), which is needed for the justification of class variable notation.

If we omit the requirement that y not occur in ph, we can derive a contradiction, as notzfaus 4840 shows. (Contributed by NM, 21-Jun-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 2690	. . . . . 6 |- ( z = A -> ( x e. z <-> x e. A ) )
3	2	anbi1d 741	. . . . 5 |- ( z = A -> ( ( x e. z /\ ph ) <-> ( x e. A /\ ph ) ) )
4	3	bibi2d 332	. . . 4 |- ( z = A -> ( ( x e. y <-> ( x e. z /\ ph ) ) <-> ( x e. y <-> ( x e. A /\ ph ) ) ) )
5	4	albidv 1849	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
6	5	exbidv 1850	. 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 4781	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8	1, 6, 7	vtocl 3259	1 |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  inex1  4799