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Theorem mobid 1978
Description: Formula-building rule for "at most one" quantifier (deduction rule). (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 1548 . . 3  |-  ( ph  ->  ( E. x ps  <->  E. x ch ) )
41, 2eubid 1950 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
53, 4imbi12d 232 . 2  |-  ( ph  ->  ( ( E. x ps  ->  E! x ps )  <->  ( E. x ch  ->  E! x ch ) ) )
6 df-mo 1947 . 2  |-  ( E* x ps  <->  ( E. x ps  ->  E! x ps ) )
7 df-mo 1947 . 2  |-  ( E* x ch  <->  ( E. x ch  ->  E! x ch ) )
85, 6, 73bitr4g 221 1  |-  ( ph  ->  ( E* x ps  <->  E* x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1390   E.wex 1422   E!weu 1943   E*wmo 1944
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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-eu 1946  df-mo 1947
This theorem is referenced by:  mobidv  1979  rmobida  2545  rmoeq1f  2553
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