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Theorem zfauscl 2695
Description: Separation Scheme using a class variable. To derive this from ax-sep 2693, we invoke the Axiom of Extensionality (indirectly via vtocl 1833), 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 2731 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
StepHypRef Expression
1 zfauscl.1 . 2 |- A e. V
2 eleq2 1527 . . . . . 6 |- (z = A -> (x e. z <-> x e. A))
32anbi1d 615 . . . . 5 |- (z = A -> ((x e. z /\ ph) <-> (x e. A /\ ph)))
43bibi2d 616 . . . 4 |- (z = A -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. A /\ ph))))
54albidv 1273 . . 3 |- (z = A -> (A.x(x e. y <-> (x e. z /\ ph)) <-> A.x(x e. y <-> (x e. A /\ ph))))
65exbidv 1274 . 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 2693 . 2 |- E.yA.x(x e. y <-> (x e. z /\ ph))
81, 6, 7vtocl 1833 1 |- E.yA.x(x e. y <-> (x e. A /\ ph))
Colors of variables: wff set class
Syntax hints:   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802
This theorem is referenced by:  nalset 2702  inex1 2706
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452  ax-sep 2693
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803
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