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Theorem bnj89 28424
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj89.1  |-  Z  e. 
_V
Assertion
Ref Expression
bnj89  |-  ( [. Z  /  y ]. E! x ph  <->  E! x [. Z  /  y ]. ph )
Distinct variable groups:    x, Z    x, y
Allowed substitution hints:    ph( x, y)    Z( y)

Proof of Theorem bnj89
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 sbcex2 3153 . . 3  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w [. Z  /  y ]. A. x ( ph  <->  x  =  w ) )
2 sbcal 3151 . . . 4  |-  ( [. Z  /  y ]. A. x ( ph  <->  x  =  w )  <->  A. x [. Z  /  y ]. ( ph  <->  x  =  w ) )
32exbii 1589 . . 3  |-  ( E. w [. Z  / 
y ]. A. x (
ph 
<->  x  =  w )  <->  E. w A. x [. Z  /  y ]. ( ph 
<->  x  =  w ) )
4 bnj89.1 . . . . . . 7  |-  Z  e. 
_V
5 sbcbig 3150 . . . . . . 7  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. ( ph  <->  x  =  w )  <->  ( [. Z  /  y ]. ph  <->  [. Z  / 
y ]. x  =  w ) ) )
64, 5ax-mp 8 . . . . . 6  |-  ( [. Z  /  y ]. ( ph 
<->  x  =  w )  <-> 
( [. Z  /  y ]. ph  <->  [. Z  /  y ]. x  =  w
) )
7 sbcg 3169 . . . . . . . 8  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. x  =  w  <->  x  =  w ) )
84, 7ax-mp 8 . . . . . . 7  |-  ( [. Z  /  y ]. x  =  w  <->  x  =  w
)
98bibi2i 305 . . . . . 6  |-  ( (
[. Z  /  y ]. ph  <->  [. Z  /  y ]. x  =  w
)  <->  ( [. Z  /  y ]. ph  <->  x  =  w ) )
106, 9bitri 241 . . . . 5  |-  ( [. Z  /  y ]. ( ph 
<->  x  =  w )  <-> 
( [. Z  /  y ]. ph  <->  x  =  w
) )
1110albii 1572 . . . 4  |-  ( A. x [. Z  /  y ]. ( ph  <->  x  =  w )  <->  A. x
( [. Z  /  y ]. ph  <->  x  =  w
) )
1211exbii 1589 . . 3  |-  ( E. w A. x [. Z  /  y ]. ( ph 
<->  x  =  w )  <->  E. w A. x (
[. Z  /  y ]. ph  <->  x  =  w
) )
131, 3, 123bitri 263 . 2  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w A. x ( [. Z  /  y ]. ph  <->  x  =  w ) )
14 df-eu 2242 . . 3  |-  ( E! x ph  <->  E. w A. x ( ph  <->  x  =  w ) )
1514sbcbii 3159 . 2  |-  ( [. Z  /  y ]. E! x ph  <->  [. Z  /  y ]. E. w A. x
( ph  <->  x  =  w
) )
16 df-eu 2242 . 2  |-  ( E! x [. Z  / 
y ]. ph  <->  E. w A. x ( [. Z  /  y ]. ph  <->  x  =  w ) )
1713, 15, 163bitr4i 269 1  |-  ( [. Z  /  y ]. E! x ph  <->  E! x [. Z  /  y ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1546   E.wex 1547    e. wcel 1717   E!weu 2238   _Vcvv 2899   [.wsbc 3104
This theorem is referenced by:  bnj130  28583  bnj207  28590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
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