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Theorem sb8mo 2301
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 2220 . . 3  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
31sb8eu 2300 . . 3  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
42, 3imbi12i 328 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
5 df-mo 2271 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
6 df-mo 2271 . 2  |-  ( E* y [ y  /  x ] ph  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
74, 5, 63bitr4i 281 1  |-  ( E* x ph  <->  E* y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   E.wex 1660   F/wnf 1664   [wsb 1787   E!weu 2266   E*wmo 2267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271
This theorem is referenced by: (None)
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