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Theorem rexbiia 2472
Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
Hypothesis
Ref Expression
ralbiia.1  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexbiia  |-  ( E. x  e.  A  ph  <->  E. x  e.  A  ps )

Proof of Theorem rexbiia
StepHypRef Expression
1 ralbiia.1 . . 3  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
21pm5.32i 450 . 2  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32rexbii2 2468 1  |-  ( E. x  e.  A  ph  <->  E. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 2128   E.wrex 2436
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-rex 2441
This theorem is referenced by:  2rexbiia  2473  ceqsrexbv  2843  reu8  2908  reldm  6136  djur  7015  prarloclem3  7419  suplocexprlemell  7635  recexgt0  8459  fsum3  11295  fprodseq  11491  even2n  11777  lmres  12718  reeff1o  13164  ioocosf1o  13245
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