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Theorem reubida 2827
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 623 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3eubid 2239 . 2  |-  ( ph  ->  ( E! x ( x  e.  A  /\  ps )  <->  E! x ( x  e.  A  /\  ch ) ) )
5 df-reu 2650 . 2  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
6 df-reu 2650 . 2  |-  ( E! x  e.  A  ch  <->  E! x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 280 1  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1550    e. wcel 1717   E!weu 2232   E!wreu 2645
This theorem is referenced by:  reubidva  2828  reuan  27620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-eu 2236  df-reu 2650
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