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Theorem eubidh 1949
Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
Hypotheses
Ref Expression
eubidh.1  |-  ( ph  ->  A. x ph )
eubidh.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
eubidh  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )

Proof of Theorem eubidh
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eubidh.1 . . . 4  |-  ( ph  ->  A. x ph )
2 eubidh.2 . . . . 5  |-  ( ph  ->  ( ps  <->  ch )
)
32bibi1d 231 . . . 4  |-  ( ph  ->  ( ( ps  <->  x  =  y )  <->  ( ch  <->  x  =  y ) ) )
41, 3albidh 1410 . . 3  |-  ( ph  ->  ( A. x ( ps  <->  x  =  y
)  <->  A. x ( ch  <->  x  =  y ) ) )
54exbidv 1748 . 2  |-  ( ph  ->  ( E. y A. x ( ps  <->  x  =  y )  <->  E. y A. x ( ch  <->  x  =  y ) ) )
6 df-eu 1946 . 2  |-  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y ) )
7 df-eu 1946 . 2  |-  ( E! x ch  <->  E. y A. x ( ch  <->  x  =  y ) )
85, 6, 73bitr4g 221 1  |-  ( ph  ->  ( E! x ps  <->  E! x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   E.wex 1422   E!weu 1943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-eu 1946
This theorem is referenced by:  euor  1969  mobidh  1977  euan  1999  euor2  2001  eupickbi  2025
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