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Theorem rexeqi 2754
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 2750 . 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 1632   E.wrex 2557
This theorem is referenced by:  rexrab2  2946  rexprg  3696  rextpg  3698  rexxp  4844  oarec  6576  4sqlem12  13019  cmpfi  17151  txbas  17278  xkobval  17297  imasdsf1olem  17953  xpsdsval  17961  plyun0  19595  coeeu  19623  1cubr  20154  adjbdln  22679  elunirnmbfm  23573  nofulllem5  24431  filnetlem4  26433  rexrabdioph  26978  fnwe2lem2  27251
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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