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Theorem eubid 1907
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 222 . . . 4  |-  ( ph  ->  ( ( ps  <->  x  =  y )  <->  ( ch  <->  x  =  y ) ) )
41, 3albid 1506 . . 3  |-  ( ph  ->  ( A. x ( ps  <->  x  =  y
)  <->  A. x ( ch  <->  x  =  y ) ) )
54exbidv 1706 . 2  |-  ( ph  ->  ( E. y A. x ( ps  <->  x  =  y )  <->  E. y A. x ( ch  <->  x  =  y ) ) )
6 df-eu 1903 . 2  |-  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y ) )
7 df-eu 1903 . 2  |-  ( E! x ch  <->  E. y A. x ( ch  <->  x  =  y ) )
85, 6, 73bitr4g 212 1  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381   E!weu 1900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-eu 1903
This theorem is referenced by:  eubidv  1908  mobid  1935  reubida  2488  reueq1f  2500  eusv2i  4183
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