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Theorem sb8eu 2161
Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
sb8eu.1  |-  F/ y
ph
Assertion
Ref Expression
sb8eu  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )

Proof of Theorem sb8eu
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1605 . . . . 5  |-  F/ w
( ph  <->  x  =  z
)
21sb8 2032 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 sbbi 2011 . . . . . 6  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z ) )
4 sb8eu.1 . . . . . . . 8  |-  F/ y
ph
54nfsb 2048 . . . . . . 7  |-  F/ y [ w  /  x ] ph
6 equsb3 2041 . . . . . . . 8  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
7 nfv 1605 . . . . . . . 8  |-  F/ y  w  =  z
86, 7nfxfr 1557 . . . . . . 7  |-  F/ y [ w  /  x ] x  =  z
95, 8nfbi 1772 . . . . . 6  |-  F/ y ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z )
103, 9nfxfr 1557 . . . . 5  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
11 nfv 1605 . . . . 5  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
12 sbequ 2000 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
1310, 11, 12cbval 1924 . . . 4  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
14 equsb3 2041 . . . . . 6  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1514sblbis 2012 . . . . 5  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1615albii 1553 . . . 4  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
172, 13, 163bitri 262 . . 3  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1817exbii 1569 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
19 df-eu 2147 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
20 df-eu 2147 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
2118, 19, 203bitr4i 268 1  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   E.wex 1528   F/wnf 1531    = wceq 1623   [wsb 1629   E!weu 2143
This theorem is referenced by:  sb8mo  2162  cbveu  2163  eu1  2164  cbvreu  2762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147
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