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Theorem zfauscl 3905
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3903, we invoke the Axiom of Extensionality (indirectly via vtocl 2625), which is needed for the justification of class variable notation. (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 2117 . . . . . 6  |-  ( z  =  A  ->  (
x  e.  z  <->  x  e.  A ) )
32anbi1d 446 . . . . 5  |-  ( z  =  A  ->  (
( x  e.  z  /\  ph )  <->  ( x  e.  A  /\  ph )
) )
43bibi2d 225 . . . 4  |-  ( z  =  A  ->  (
( x  e.  y  <-> 
( x  e.  z  /\  ph ) )  <-> 
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) ) )
54albidv 1721 . . 3  |-  ( z  =  A  ->  ( A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)  <->  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
) ) )
65exbidv 1722 . 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 3903 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
81, 6, 7vtocl 2625 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  inex1  3919  bj-d0clsepcl  10436
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