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Theorem rexeqi 2910
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 2906 . 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 178    = wceq 1653   E.wrex 2707
This theorem is referenced by:  rexrab2  3103  rexprg  3859  rextpg  3861  rexxp  5018  oarec  6806  4sqlem12  13325  cmpfi  17472  txbas  17600  xkobval  17619  ustn0  18251  imasdsf1olem  18404  xpsdsval  18412  plyun0  20117  coeeu  20145  1cubr  20683  nbgraf1olem1  21452  adjbdln  23587  elunirnmbfm  24604  nofulllem5  25662  filnetlem4  26411  rexrabdioph  26855  fnwe2lem2  27127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712
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