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Theorem sb8eu 1990
Description: Variable substitution in unique existential 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 1493 . . . . 5  |-  F/ w
( ph  <->  x  =  z
)
21sb8 1812 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 sbbi 1910 . . . . . 6  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z ) )
4 sb8eu.1 . . . . . . . 8  |-  F/ y
ph
54nfsb 1899 . . . . . . 7  |-  F/ y [ w  /  x ] ph
6 equsb3 1902 . . . . . . . 8  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
7 nfv 1493 . . . . . . . 8  |-  F/ y  w  =  z
86, 7nfxfr 1435 . . . . . . 7  |-  F/ y [ w  /  x ] x  =  z
95, 8nfbi 1553 . . . . . 6  |-  F/ y ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z )
103, 9nfxfr 1435 . . . . 5  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
11 nfv 1493 . . . . 5  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
12 sbequ 1796 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
1310, 11, 12cbval 1712 . . . 4  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
14 equsb3 1902 . . . . . 6  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1514sblbis 1911 . . . . 5  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1615albii 1431 . . . 4  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
172, 13, 163bitri 205 . . 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 1980 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
20 df-eu 1980 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
2118, 19, 203bitr4i 211 1  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1314   F/wnf 1421   E.wex 1453   [wsb 1720   E!weu 1977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980
This theorem is referenced by:  sb8mo  1991  nfeud  1993  nfeu  1996  cbveu  2001  cbvreu  2629  acexmid  5741
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