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Theorem reubiia 2618
Description: Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
Hypothesis
Ref Expression
reubiia.1  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
reubiia  |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )

Proof of Theorem reubiia
StepHypRef Expression
1 reubiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
21pm5.32i 450 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32eubii 2009 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! x ( x  e.  A  /\  ps )
)
4 df-reu 2424 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
5 df-reu 2424 . 2  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
63, 4, 53bitr4i 211 1  |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1481   E!weu 2000   E!wreu 2419
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-eu 2003  df-reu 2424
This theorem is referenced by:  reubii  2619
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