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Theorem axsep2 4142
Description: A less restrictive version of the Separation Scheme axsep 4140, where variables  x and  z can both appear free in the wff  ph, which can therefore be thought of as  ph ( x ,  z ). This version was derived from the more restrictive ax-sep 4141 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
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 axsep2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eleq2 2344 . . . . . . 7  |-  ( w  =  z  ->  (
x  e.  w  <->  x  e.  z ) )
21anbi1d 685 . . . . . 6  |-  ( w  =  z  ->  (
( x  e.  w  /\  ( x  e.  z  /\  ph ) )  <-> 
( x  e.  z  /\  ( x  e.  z  /\  ph )
) ) )
3 anabs5 784 . . . . . 6  |-  ( ( x  e.  z  /\  ( x  e.  z  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
42, 3syl6bb 252 . . . . 5  |-  ( w  =  z  ->  (
( x  e.  w  /\  ( x  e.  z  /\  ph ) )  <-> 
( x  e.  z  /\  ph ) ) )
54bibi2d 309 . . . 4  |-  ( w  =  z  ->  (
( x  e.  y  <-> 
( x  e.  w  /\  ( x  e.  z  /\  ph ) ) )  <->  ( x  e.  y  <->  ( x  e.  z  /\  ph )
) ) )
65albidv 1611 . . 3  |-  ( w  =  z  ->  ( A. x ( x  e.  y  <->  ( x  e.  w  /\  ( x  e.  z  /\  ph ) ) )  <->  A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) ) ) )
76exbidv 1612 . 2  |-  ( w  =  z  ->  ( E. y A. x ( x  e.  y  <->  ( x  e.  w  /\  (
x  e.  z  /\  ph ) ) )  <->  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
) ) )
8 ax-sep 4141 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  w  /\  ( x  e.  z  /\  ph ) ) )
97, 8chvarv 1953 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279
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