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Theorem rexeqbi1dv 2758
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 2750 . 2  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
2 raleqd.1 . . 3  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
32rexbidv 2577 . 2  |-  ( A  =  B  ->  ( E. x  e.  B  ph  <->  E. x  e.  B  ps ) )
41, 3bitrd 244 1  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632   E.wrex 2557
This theorem is referenced by:  fri  4371  frsn  4776  isofrlem  5853  f1oweALT  5867  frxp  6241  1sdom  7081  oieq2  7244  zfregcl  7324  ishaus  17066  isreg  17076  isnrm  17079  lebnumlem3  18477  isgrpo  20879  isexid2  21008  ismndo2  21028  rngomndo  21104  pjhth  21988  frmin  24313  tcnvec  25793  isibg2  26213  stoweidlem28  27880  bnj1154  29345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562
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