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Theorem reubiia 3021
Description: Formula-building rule for restricted existential quantifier (inference rule). (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 641 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32eubii 2290 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! x ( x  e.  A  /\  ps )
)
4 df-reu 2789 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
5 df-reu 2789 . 2  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
63, 4, 53bitr4i 280 1  |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    e. wcel 1870   E!weu 2266   E!wreu 2784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-12 1907
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-eu 2270  df-reu 2789
This theorem is referenced by:  reubii  3022  riotaxfrd  6297  infempty  8022
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