Theorem zfauscl 2779

Description: Separation Scheme using a class variable. To derive this from ax-sep 2777, we invoke the Axiom of Extensionality (indirectly via vtocl 1888), 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 2815 shows.

Hypothesis

Ref	Expression
zfauscl.1	|- A e. V

Assertion

Ref	Expression
zfauscl	|- E.yA.x(x e. y <-> (x e. A /\ ph))

Distinct variable groups:   x,y,A   ph,y

Proof of Theorem zfauscl

Step	Hyp	Ref	Expression
1		zfauscl.1	. 2 |- A e. V
2		eleq2 1578	. . . . . 6 |- (z = A -> (x e. z <-> x e. A))
3	2	anbi1d 620	. . . . 5 |- (z = A -> ((x e. z /\ ph) <-> (x e. A /\ ph)))
4	3	bibi2d 621	. . . 4 |- (z = A -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. A /\ ph))))
5	4	albidv 1316	. . 3 |- (z = A -> (A.x(x e. y <-> (x e. z /\ ph)) <-> A.x(x e. y <-> (x e. A /\ ph))))
6	5	exbidv 1317	. 2 |- (z = A -> (E.yA.x(x e. y <-> (x e. z /\ ph)) <-> E.yA.x(x e. y <-> (x e. A /\ ph))))
7		ax-sep 2777	. 2 |- E.yA.x(x e. y <-> (x e. z /\ ph))
8	1, 6, 7	vtocl 1888	1 |- E.yA.x(x e. y <-> (x e. A /\ ph))

Syntax hints:   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  E.wex 1016  Vcvv 1857

This theorem is referenced by:  nalset 2786  inex1 2790

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 999  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-ext 1500  ax-sep 2777

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858

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