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Theorem rexeqbi1dv 2913
Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
Hypothesis
Ref Expression
raleqd.1  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexeqbi1dv  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ps ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rexeqbi1dv
StepHypRef Expression
1 rexeq 2905 . 2  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
2 raleqd.1 . . 3  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
32rexbidv 2726 . 2  |-  ( A  =  B  ->  ( E. x  e.  B  ph  <->  E. x  e.  B  ps ) )
41, 3bitrd 245 1  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   E.wrex 2706
This theorem is referenced by:  fri  4544  frsn  4948  isofrlem  6060  f1oweALT  6074  frxp  6456  1sdom  7311  oieq2  7482  zfregcl  7562  ishaus  17386  isreg  17396  isnrm  17399  lebnumlem3  18988  isgrpo  21784  isexid2  21913  ismndo2  21933  rngomndo  22009  pjhth  22895  frmin  25517  stoweidlem28  27753  1vwmgra  28393  3vfriswmgra  28395  bnj1154  29368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711
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