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Theorem eubid 2256
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 317 . . . 4  |-  ( ph  ->  ( ( ps  <->  x  =  y )  <->  ( ch  <->  x  =  y ) ) )
41, 3albid 1907 . . 3  |-  ( ph  ->  ( A. x ( ps  <->  x  =  y
)  <->  A. x ( ch  <->  x  =  y ) ) )
54exbidv 1733 . 2  |-  ( ph  ->  ( E. y A. x ( ps  <->  x  =  y )  <->  E. y A. x ( ch  <->  x  =  y ) ) )
6 df-eu 2240 . 2  |-  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y ) )
7 df-eu 2240 . 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 1401   E.wex 1631   F/wnf 1635   E!weu 2236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-12 1876
This theorem depends on definitions:  df-bi 185  df-ex 1632  df-nf 1636  df-eu 2240
This theorem is referenced by:  mobid  2257  eubidv  2258  euor  2282  euor2  2284  euan  2300  reubida  2987  reueq1f  2999  eusv2i  4588  reusv2lem3  4594  eubi  36155
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