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Theorem sb8eu 1961
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 1466 . . . . 5  |-  F/ w
( ph  <->  x  =  z
)
21sb8 1784 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 sbbi 1881 . . . . . 6  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z ) )
4 sb8eu.1 . . . . . . . 8  |-  F/ y
ph
54nfsb 1870 . . . . . . 7  |-  F/ y [ w  /  x ] ph
6 equsb3 1873 . . . . . . . 8  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
7 nfv 1466 . . . . . . . 8  |-  F/ y  w  =  z
86, 7nfxfr 1408 . . . . . . 7  |-  F/ y [ w  /  x ] x  =  z
95, 8nfbi 1526 . . . . . 6  |-  F/ y ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z )
103, 9nfxfr 1408 . . . . 5  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
11 nfv 1466 . . . . 5  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
12 sbequ 1768 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
1310, 11, 12cbval 1684 . . . 4  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
14 equsb3 1873 . . . . . 6  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1514sblbis 1882 . . . . 5  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1615albii 1404 . . . 4  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
172, 13, 163bitri 204 . . 3  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1817exbii 1541 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
19 df-eu 1951 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
20 df-eu 1951 . 2  |-  ( E! y [ y  /  x ] ph  <->  E. z A. y ( [ y  /  x ] ph  <->  y  =  z ) )
2118, 19, 203bitr4i 210 1  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1287   F/wnf 1394   E.wex 1426   [wsb 1692   E!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by:  sb8mo  1962  nfeud  1964  nfeu  1967  cbveu  1972  cbvreu  2588  acexmid  5633
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