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Theorem bnj89 28820
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 bnj89.1 . . . 4  |-  Z  e. 
_V
2 sbcexg 3043 . . . 4  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. E. w A. x
( ph  <->  x  =  w
)  <->  E. w [. Z  /  y ]. A. x ( ph  <->  x  =  w ) ) )
31, 2ax-mp 8 . . 3  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w [. Z  /  y ]. A. x ( ph  <->  x  =  w ) )
4 sbcalg 3041 . . . . 5  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. A. x ( ph  <->  x  =  w )  <->  A. x [. Z  /  y ]. ( ph  <->  x  =  w ) ) )
51, 4ax-mp 8 . . . 4  |-  ( [. Z  /  y ]. A. x ( ph  <->  x  =  w )  <->  A. x [. Z  /  y ]. ( ph  <->  x  =  w ) )
65exbii 1571 . . 3  |-  ( E. w [. Z  / 
y ]. A. x (
ph 
<->  x  =  w )  <->  E. w A. x [. Z  /  y ]. ( ph 
<->  x  =  w ) )
7 sbcbig 3039 . . . . . . 7  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. ( ph  <->  x  =  w )  <->  ( [. Z  /  y ]. ph  <->  [. Z  / 
y ]. x  =  w ) ) )
81, 7ax-mp 8 . . . . . 6  |-  ( [. Z  /  y ]. ( ph 
<->  x  =  w )  <-> 
( [. Z  /  y ]. ph  <->  [. Z  /  y ]. x  =  w
) )
9 sbcg 3058 . . . . . . . 8  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. x  =  w  <->  x  =  w ) )
101, 9ax-mp 8 . . . . . . 7  |-  ( [. Z  /  y ]. x  =  w  <->  x  =  w
)
1110bibi2i 304 . . . . . 6  |-  ( (
[. Z  /  y ]. ph  <->  [. Z  /  y ]. x  =  w
)  <->  ( [. Z  /  y ]. ph  <->  x  =  w ) )
128, 11bitri 240 . . . . 5  |-  ( [. Z  /  y ]. ( ph 
<->  x  =  w )  <-> 
( [. Z  /  y ]. ph  <->  x  =  w
) )
1312albii 1555 . . . 4  |-  ( A. x [. Z  /  y ]. ( ph  <->  x  =  w )  <->  A. x
( [. Z  /  y ]. ph  <->  x  =  w
) )
1413exbii 1571 . . 3  |-  ( E. w A. x [. Z  /  y ]. ( ph 
<->  x  =  w )  <->  E. w A. x (
[. Z  /  y ]. ph  <->  x  =  w
) )
153, 6, 143bitri 262 . 2  |-  ( [. Z  /  y ]. E. w A. x ( ph  <->  x  =  w )  <->  E. w A. x ( [. Z  /  y ]. ph  <->  x  =  w ) )
16 df-eu 2149 . . . 4  |-  ( E! x ph  <->  E. w A. x ( ph  <->  x  =  w ) )
1716sbcbiiOLD 3049 . . 3  |-  ( Z  e.  _V  ->  ( [. Z  /  y ]. E! x ph  <->  [. Z  / 
y ]. E. w A. x ( ph  <->  x  =  w ) ) )
181, 17ax-mp 8 . 2  |-  ( [. Z  /  y ]. E! x ph  <->  [. Z  /  y ]. E. w A. x
( ph  <->  x  =  w
) )
19 df-eu 2149 . 2  |-  ( E! x [. Z  / 
y ]. ph  <->  E. w A. x ( [. Z  /  y ]. ph  <->  x  =  w ) )
2015, 18, 193bitr4i 268 1  |-  ( [. Z  /  y ]. E! x ph  <->  E! x [. Z  /  y ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1529   E.wex 1530    = wceq 1625    e. wcel 1686   E!weu 2145   _Vcvv 2790   [.wsbc 2993
This theorem is referenced by:  bnj130  28979  bnj207  28986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-sbc 2994
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