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Theorem eubid 2289
Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
Hypotheses
Ref Expression
eubid.1  |-  F/ x ph
eubid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
eubid  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )

Proof of Theorem eubid
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eubid.1 . . . 4  |-  F/ x ph
2 eubid.2 . . . . 5  |-  ( ph  ->  ( ps  <->  ch )
)
32bibi1d 319 . . . 4  |-  ( ph  ->  ( ( ps  <->  x  =  y )  <->  ( ch  <->  x  =  y ) ) )
41, 3albid 1828 . . 3  |-  ( ph  ->  ( A. x ( ps  <->  x  =  y
)  <->  A. x ( ch  <->  x  =  y ) ) )
54exbidv 1685 . 2  |-  ( ph  ->  ( E. y A. x ( ps  <->  x  =  y )  <->  E. y A. x ( ch  <->  x  =  y ) ) )
6 df-eu 2272 . 2  |-  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y ) )
7 df-eu 2272 . 2  |-  ( E! x ch  <->  E. y A. x ( ch  <->  x  =  y ) )
85, 6, 73bitr4g 288 1  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1372   E.wex 1591   F/wnf 1594   E!weu 2268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-12 1798
This theorem depends on definitions:  df-bi 185  df-ex 1592  df-nf 1595  df-eu 2272
This theorem is referenced by:  mobid  2290  eubidv  2291  euor  2325  euor2  2327  euor2OLD  2328  euan  2346  euanOLD  2347  eupickbiOLD  2362  reubida  3037  reueq1f  3049  eusv2i  4637  reusv2lem3  4643  eubi  30876
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