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Theorem bnj89 33071
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 3385 . . 3  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w [. Z  /  y ]. A. x ( ph  <->  x  =  w ) )
2 sbcal 3383 . . . 4  |-  ( [. Z  /  y ]. A. x ( ph  <->  x  =  w )  <->  A. x [. Z  /  y ]. ( ph  <->  x  =  w ) )
32exbii 1644 . . 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 3378 . . . . . . 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 3405 . . . . . . . 8  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. x  =  w  <->  x  =  w ) )
84, 7ax-mp 5 . . . . . . 7  |-  ( [. Z  /  y ]. x  =  w  <->  x  =  w
)
98bibi2i 313 . . . . . 6  |-  ( (
[. Z  /  y ]. ph  <->  [. Z  /  y ]. x  =  w
)  <->  ( [. Z  /  y ]. ph  <->  x  =  w ) )
106, 9bitri 249 . . . . 5  |-  ( [. Z  /  y ]. ( ph 
<->  x  =  w )  <-> 
( [. Z  /  y ]. ph  <->  x  =  w
) )
1110albii 1620 . . . 4  |-  ( A. x [. Z  /  y ]. ( ph  <->  x  =  w )  <->  A. x
( [. Z  /  y ]. ph  <->  x  =  w
) )
1211exbii 1644 . . 3  |-  ( E. w A. x [. Z  /  y ]. ( ph 
<->  x  =  w )  <->  E. w A. x (
[. Z  /  y ]. ph  <->  x  =  w
) )
131, 3, 123bitri 271 . 2  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w A. x ( [. Z  /  y ]. ph  <->  x  =  w ) )
14 df-eu 2279 . . 3  |-  ( E! x ph  <->  E. w A. x ( ph  <->  x  =  w ) )
1514sbcbii 3391 . 2  |-  ( [. Z  /  y ]. E! x ph  <->  [. Z  /  y ]. E. w A. x
( ph  <->  x  =  w
) )
16 df-eu 2279 . 2  |-  ( E! x [. Z  / 
y ]. ph  <->  E. w A. x ( [. Z  /  y ]. ph  <->  x  =  w ) )
1713, 15, 163bitr4i 277 1  |-  ( [. Z  /  y ]. E! x ph  <->  E! x [. Z  /  y ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1377   E.wex 1596    e. wcel 1767   E!weu 2275   _Vcvv 3113   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  bnj130  33228  bnj207  33235
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