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Theorem eubid 2013
Description: Formula-building rule for unique existential quantifier (deduction form). (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 232 . . . 4  |-  ( ph  ->  ( ( ps  <->  x  =  y )  <->  ( ch  <->  x  =  y ) ) )
41, 3albid 1595 . . 3  |-  ( ph  ->  ( A. x ( ps  <->  x  =  y
)  <->  A. x ( ch  <->  x  =  y ) ) )
54exbidv 1805 . 2  |-  ( ph  ->  ( E. y A. x ( ps  <->  x  =  y )  <->  E. y A. x ( ch  <->  x  =  y ) ) )
6 df-eu 2009 . 2  |-  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y ) )
7 df-eu 2009 . 2  |-  ( E! x ch  <->  E. y A. x ( ch  <->  x  =  y ) )
85, 6, 73bitr4g 222 1  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1333   F/wnf 1440   E.wex 1472   E!weu 2006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-eu 2009
This theorem is referenced by:  eubidv  2014  mobid  2041  reubida  2638  reueq1f  2650  eusv2i  4415
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