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Theorem sb8mo 1962
Description: Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
sb8eu.1  |-  F/ y
ph
Assertion
Ref Expression
sb8mo  |-  ( E* x ph  <->  E* y [ y  /  x ] ph )

Proof of Theorem sb8mo
StepHypRef Expression
1 sb8eu.1 . . . 4  |-  F/ y
ph
21sb8e 1785 . . 3  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
31sb8eu 1961 . . 3  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
42, 3imbi12i 237 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
5 df-mo 1952 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
6 df-mo 1952 . 2  |-  ( E* y [ y  /  x ] ph  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
74, 5, 63bitr4i 210 1  |-  ( E* x ph  <->  E* y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1394   E.wex 1426   [wsb 1692   E!weu 1948   E*wmo 1949
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  df-mo 1952
This theorem is referenced by: (None)
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