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Theorem sb8mo 2300
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 2169 . . 3  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
31sb8eu 2299 . . 3  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
42, 3imbi12i 317 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
5 df-mo 2286 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
6 df-mo 2286 . 2  |-  ( E* y [ y  /  x ] ph  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
74, 5, 63bitr4i 269 1  |-  ( E* x ph  <->  E* y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1550   F/wnf 1553   [wsb 1658   E!weu 2281   E*wmo 2282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286
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