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Theorem zfauscl 4159
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4157, we invoke the Axiom of Extensionality (indirectly via vtocl 2851), 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 4201 shows. (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.
StepHypRef Expression
1 zfauscl.1 . 2  |-  A  e. 
_V
2 eleq2 2357 . . . . . 6  |-  ( z  =  A  ->  (
x  e.  z  <->  x  e.  A ) )
32anbi1d 685 . . . . 5  |-  ( z  =  A  ->  (
( x  e.  z  /\  ph )  <->  ( x  e.  A  /\  ph )
) )
43bibi2d 309 . . . 4  |-  ( z  =  A  ->  (
( x  e.  y  <-> 
( x  e.  z  /\  ph ) )  <-> 
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) ) )
54albidv 1615 . . 3  |-  ( z  =  A  ->  ( A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)  <->  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
) ) )
65exbidv 1616 . 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 4157 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
81, 6, 7vtocl 2851 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  inex1  4171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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