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Theorem mobid 2010
Description: Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
Hypotheses
Ref Expression
mobid.1  |-  F/ x ph
mobid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
mobid  |-  ( ph  ->  ( E* x ps  <->  E* x ch ) )

Proof of Theorem mobid
StepHypRef Expression
1 mobid.1 . . . 4  |-  F/ x ph
2 mobid.2 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2exbid 1578 . . 3  |-  ( ph  ->  ( E. x ps  <->  E. x ch ) )
41, 2eubid 1982 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
53, 4imbi12d 233 . 2  |-  ( ph  ->  ( ( E. x ps  ->  E! x ps )  <->  ( E. x ch  ->  E! x ch ) ) )
6 df-mo 1979 . 2  |-  ( E* x ps  <->  ( E. x ps  ->  E! x ps ) )
7 df-mo 1979 . 2  |-  ( E* x ch  <->  ( E. x ch  ->  E! x ch ) )
85, 6, 73bitr4g 222 1  |-  ( ph  ->  ( E* x ps  <->  E* x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1419   E.wex 1451   E!weu 1975   E*wmo 1976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-eu 1978  df-mo 1979
This theorem is referenced by:  mobidv  2011  rmobida  2591  rmoeq1f  2599
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