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Theorem rexeqbi1dv 2745
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 2737 . 2  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
2 raleqd.1 . . 3  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
32rexbidv 2564 . 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 1623   E.wrex 2544
This theorem is referenced by:  fri  4355  frsn  4760  isofrlem  5837  f1oweALT  5851  frxp  6225  1sdom  7065  oieq2  7228  zfregcl  7308  ishaus  17050  isreg  17060  isnrm  17063  lebnumlem3  18461  isgrpo  20863  isexid2  20992  ismndo2  21012  rngomndo  21088  pjhth  21972  frmin  24242  tcnvec  25690  isibg2  26110  stoweidlem28  27777  bnj1154  29029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549
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