Theorem zfauscl 2710

Description: Separation Scheme using a class variable. To derive this from ax-sep 2708, we invoke the Axiom of Extensionality (indirectly via vtocl 1845), 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 2746 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 1538	. . . . . 6 |- (z = A -> (x e. z <-> x e. A))
3	2	anbi1d 619	. . . . 5 |- (z = A -> ((x e. z /\ ph) <-> (x e. A /\ ph)))
4	3	bibi2d 620	. . . 4 |- (z = A -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. A /\ ph))))
5	4	albidv 1280	. . 3 |- (z = A -> (A.x(x e. y <-> (x e. z /\ ph)) <-> A.x(x e. y <-> (x e. A /\ ph))))
6	5	exbidv 1281	. 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 2708	. 2 |- E.yA.x(x e. y <-> (x e. z /\ ph))
8	1, 6, 7	vtocl 1845	1 |- E.yA.x(x e. y <-> (x e. A /\ ph))

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814

This theorem is referenced by:  nalset 2717  inex1 2721

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462  ax-sep 2708

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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