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Theorem eubid 2290
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 312 . . . 4  |-  ( ph  ->  ( ( ps  <->  x  =  y )  <->  ( ch  <->  x  =  y ) ) )
41, 3albid 1789 . . 3  |-  ( ph  ->  ( A. x ( ps  <->  x  =  y
)  <->  A. x ( ch  <->  x  =  y ) ) )
54exbidv 1637 . 2  |-  ( ph  ->  ( E. y A. x ( ps  <->  x  =  y )  <->  E. y A. x ( ch  <->  x  =  y ) ) )
6 df-eu 2287 . 2  |-  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y ) )
7 df-eu 2287 . 2  |-  ( E! x ch  <->  E. y A. x ( ch  <->  x  =  y ) )
85, 6, 73bitr4g 281 1  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   E.wex 1551   F/wnf 1554   E!weu 2283
This theorem is referenced by:  eubidv  2291  euor  2310  mobid  2317  euan  2340  eupickbi  2349  euor2  2351  reubida  2892  reueq1f  2904  eusv2i  4722  reusv2lem3  4728  eubi  27615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-ex 1552  df-nf 1555  df-eu 2287
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