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Theorem mobid 2285
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 1937 . . 3  |-  ( ph  ->  ( E. x ps  <->  E. x ch ) )
41, 2eubid 2284 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
53, 4imbi12d 321 . 2  |-  ( ph  ->  ( ( E. x ps  ->  E! x ps )  <->  ( E. x ch  ->  E! x ch ) ) )
6 df-mo 2270 . 2  |-  ( E* x ps  <->  ( E. x ps  ->  E! x ps ) )
7 df-mo 2270 . 2  |-  ( E* x ch  <->  ( E. x ch  ->  E! x ch ) )
85, 6, 73bitr4g 291 1  |-  ( ph  ->  ( E* x ps  <->  E* x ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   E.wex 1659   F/wnf 1663   E!weu 2265   E*wmo 2266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-12 1905
This theorem depends on definitions:  df-bi 188  df-ex 1660  df-nf 1664  df-eu 2269  df-mo 2270
This theorem is referenced by:  mobidv  2287  moanim  2325  rmobida  3016  rmoeq1f  3024  funcnvmptOLD  28257  funcnvmpt  28258
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