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Theorem reubida 3040
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
reubida.1  |-  F/ x ph
reubida.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
reubida  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)

Proof of Theorem reubida
StepHypRef Expression
1 reubida.1 . . 3  |-  F/ x ph
2 reubida.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32pm5.32da 641 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3eubid 2303 . 2  |-  ( ph  ->  ( E! x ( x  e.  A  /\  ps )  <->  E! x ( x  e.  A  /\  ch ) ) )
5 df-reu 2814 . 2  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
6 df-reu 2814 . 2  |-  ( E! x  e.  A  ch  <->  E! x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 288 1  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   F/wnf 1617    e. wcel 1819   E!weu 2283   E!wreu 2809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-12 1855
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618  df-eu 2287  df-reu 2814
This theorem is referenced by:  reubidva  3041  reuan  32431
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