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Theorem reubiia 2759
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 618 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32eubii 2185 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! x ( x  e.  A  /\  ps )
)
4 df-reu 2584 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
5 df-reu 2584 . 2  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
63, 4, 53bitr4i 268 1  |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1701   E!weu 2176   E!wreu 2579
This theorem is referenced by:  reubii  2760  riotaxfrd  6378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-11 1732
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-eu 2180  df-reu 2584
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