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Theorem axsep2 4158
Description: A less restrictive version of the Separation Scheme axsep 4156, 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 4157 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 2357 . . . . . . 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 1615 . . 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 1616 . 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 4157 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  w  /\  ( x  e.  z  /\  ph ) ) )
97, 8chvarv 1966 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 1530   E.wex 1531    = wceq 1632    e. wcel 1696
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-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-cleq 2289  df-clel 2292
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