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Theorem bnj89 29086
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 3210 . . 3  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w [. Z  /  y ]. A. x ( ph  <->  x  =  w ) )
2 sbcal 3208 . . . 4  |-  ( [. Z  /  y ]. A. x ( ph  <->  x  =  w )  <->  A. x [. Z  /  y ]. ( ph  <->  x  =  w ) )
32exbii 1592 . . 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 3207 . . . . . . 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 3226 . . . . . . . 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 1575 . . . 4  |-  ( A. x [. Z  /  y ]. ( ph  <->  x  =  w )  <->  A. x
( [. Z  /  y ]. ph  <->  x  =  w
) )
1211exbii 1592 . . 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 2285 . . 3  |-  ( E! x ph  <->  E. w A. x ( ph  <->  x  =  w ) )
1514sbcbii 3216 . 2  |-  ( [. Z  /  y ]. E! x ph  <->  [. Z  /  y ]. E. w A. x
( ph  <->  x  =  w
) )
16 df-eu 2285 . 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 1549   E.wex 1550    e. wcel 1725   E!weu 2281   _Vcvv 2956   [.wsbc 3161
This theorem is referenced by:  bnj130  29245  bnj207  29252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162
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