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Theorem bnj89 29599
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 3306 . . 3  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w [. Z  /  y ]. A. x ( ph  <->  x  =  w ) )
2 sbcal 3305 . . . 4  |-  ( [. Z  /  y ]. A. x ( ph  <->  x  =  w )  <->  A. x [. Z  /  y ]. ( ph  <->  x  =  w ) )
32exbii 1726 . . 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 3300 . . . . . . 7  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. ( ph  <->  x  =  w )  <->  ( [. Z  /  y ]. ph  <->  [. Z  / 
y ]. x  =  w ) ) )
64, 5ax-mp 5 . . . . . 6  |-  ( [. Z  /  y ]. ( ph 
<->  x  =  w )  <-> 
( [. Z  /  y ]. ph  <->  [. Z  /  y ]. x  =  w
) )
7 sbcg 3321 . . . . . . . 8  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. x  =  w  <->  x  =  w ) )
84, 7ax-mp 5 . . . . . . 7  |-  ( [. Z  /  y ]. x  =  w  <->  x  =  w
)
98bibi2i 320 . . . . . 6  |-  ( (
[. Z  /  y ]. ph  <->  [. Z  /  y ]. x  =  w
)  <->  ( [. Z  /  y ]. ph  <->  x  =  w ) )
106, 9bitri 257 . . . . 5  |-  ( [. Z  /  y ]. ( ph 
<->  x  =  w )  <-> 
( [. Z  /  y ]. ph  <->  x  =  w
) )
1110albii 1699 . . . 4  |-  ( A. x [. Z  /  y ]. ( ph  <->  x  =  w )  <->  A. x
( [. Z  /  y ]. ph  <->  x  =  w
) )
1211exbii 1726 . . 3  |-  ( E. w A. x [. Z  /  y ]. ( ph 
<->  x  =  w )  <->  E. w A. x (
[. Z  /  y ]. ph  <->  x  =  w
) )
131, 3, 123bitri 279 . 2  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w A. x ( [. Z  /  y ]. ph  <->  x  =  w ) )
14 df-eu 2323 . . 3  |-  ( E! x ph  <->  E. w A. x ( ph  <->  x  =  w ) )
1514sbcbii 3311 . 2  |-  ( [. Z  /  y ]. E! x ph  <->  [. Z  /  y ]. E. w A. x
( ph  <->  x  =  w
) )
16 df-eu 2323 . 2  |-  ( E! x [. Z  / 
y ]. ph  <->  E. w A. x ( [. Z  /  y ]. ph  <->  x  =  w ) )
1713, 15, 163bitr4i 285 1  |-  ( [. Z  /  y ]. E! x ph  <->  E! x [. Z  /  y ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1450   E.wex 1671    e. wcel 1904   E!weu 2319   _Vcvv 3031   [.wsbc 3255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033  df-sbc 3256
This theorem is referenced by:  bnj130  29757  bnj207  29764
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