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Theorem eubidh 1954
Description: Formula-building rule for unique existential quantifier (deduction form). (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 1414 . . 3  |-  ( ph  ->  ( A. x ( ps  <->  x  =  y
)  <->  A. x ( ch  <->  x  =  y ) ) )
54exbidv 1753 . 2  |-  ( ph  ->  ( E. y A. x ( ps  <->  x  =  y )  <->  E. y A. x ( ch  <->  x  =  y ) ) )
6 df-eu 1951 . 2  |-  ( E! x ps  <->  E. y A. x ( ps  <->  x  =  y ) )
7 df-eu 1951 . 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 1287   E.wex 1426   E!weu 1948
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-eu 1951
This theorem is referenced by:  euor  1974  mobidh  1982  euan  2004  euor2  2006  eupickbi  2030
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