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Theorem bj-axsep2 34894
Description: Remove dependency on ax-13 2004, ax-ext 2432, df-cleq 2446 and df-clel 2449 from axsep2 4561 while shortening its proof. Remark: the comment in zfauscl 4562 is misleading: the essential use of ax-ext 2432 is the one via eleq2 2527 and not the one via vtocl 3158, since the latter can be proved without ax-ext 2432 (see bj-vtocl 34882). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axsep2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Distinct variable groups:    x, y,
z    ph, y
Allowed substitution hints:    ph( x, z)

Proof of Theorem bj-axsep2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elequ2 1828 . . . . . 6  |-  ( w  =  z  ->  (
x  e.  w  <->  x  e.  z ) )
21anbi1d 702 . . . . 5  |-  ( w  =  z  ->  (
( x  e.  w  /\  ph )  <->  ( x  e.  z  /\  ph )
) )
32bibi2d 316 . . . 4  |-  ( w  =  z  ->  (
( x  e.  y  <-> 
( x  e.  w  /\  ph ) )  <->  ( x  e.  y  <->  ( x  e.  z  /\  ph )
) ) )
43albidv 1718 . . 3  |-  ( w  =  z  ->  ( A. x ( x  e.  y  <->  ( x  e.  w  /\  ph )
)  <->  A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
) ) )
54exbidv 1719 . 2  |-  ( w  =  z  ->  ( E. y A. x ( x  e.  y  <->  ( x  e.  w  /\  ph )
)  <->  E. y A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) ) ) )
6 ax-sep 4560 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  w  /\  ph )
)
75, 6bj-chvarvv 34688 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367   A.wal 1396   E.wex 1617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-12 1859  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622
This theorem is referenced by: (None)
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