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Theorem rexeqi 2741
Description: Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
raleq1i.1  |-  A  =  B
Assertion
Ref Expression
rexeqi  |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem rexeqi
StepHypRef Expression
1 raleq1i.1 . 2  |-  A  =  B
2 rexeq 2737 . 2  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
31, 2ax-mp 8 1  |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   E.wrex 2544
This theorem is referenced by:  rexrab2  2933  rexprg  3683  rextpg  3685  rexxp  4828  oarec  6560  4sqlem12  13003  cmpfi  17135  txbas  17262  xkobval  17281  imasdsf1olem  17937  xpsdsval  17945  plyun0  19579  coeeu  19607  1cubr  20138  adjbdln  22663  elunirnmbfm  23558  nofulllem5  24360  filnetlem4  26330  rexrabdioph  26875  fnwe2lem2  27148
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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