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Theorem mobid 2190
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 1765 . . 3  |-  ( ph  ->  ( E. x ps  <->  E. x ch ) )
41, 2eubid 2163 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
53, 4imbi12d 311 . 2  |-  ( ph  ->  ( ( E. x ps  ->  E! x ps )  <->  ( E. x ch  ->  E! x ch ) ) )
6 df-mo 2161 . 2  |-  ( E* x ps  <->  ( E. x ps  ->  E! x ps ) )
7 df-mo 2161 . 2  |-  ( E* x ch  <->  ( E. x ch  ->  E! x ch ) )
85, 6, 73bitr4g 279 1  |-  ( ph  ->  ( E* x ps  <->  E* x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1531   F/wnf 1534   E!weu 2156   E*wmo 2157
This theorem is referenced by:  mobidv  2191  euan  2213  rmobida  2740  rmoeq1f  2748  funcnvmptOLD  23249  funcnvmpt  23250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-ex 1532  df-nf 1535  df-eu 2160  df-mo 2161
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