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Theorem rexeqi 3037
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 3033 . 2  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
31, 2ax-mp 5 1  |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437   E.wrex 2783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rex 2788
This theorem is referenced by:  rexrab2  3245  rexprg  4053  rextpg  4055  rexxp  4997  oarec  7271  wwlktovfo  13012  4sqlem12  14863  pmatcollpw3fi1  19743  cmpfi  20354  txbas  20513  xkobval  20532  ustn0  21166  imasdsf1olem  21319  xpsdsval  21327  plyun0  23019  coeeu  23047  1cubr  23633  nbgraf1olem1  25014  wlknwwlknsur  25285  wlkiswwlksur  25292  adjbdln  27571  elunirnmbfm  28914  nofulllem5  30380  filnetlem4  30822  rexrabdioph  35346  fnwe2lem2  35615  fourierdlem70  37608  fourierdlem80  37618
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