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Theorem rexeq 2604
Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
Assertion
Ref Expression
rexeq  |-  ( A  =  B  ->  ( 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 rexeq
StepHypRef Expression
1 nfcv 2258 . 2  |-  F/_ x A
2 nfcv 2258 . 2  |-  F/_ x B
31, 2rexeqf 2600 1  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316   E.wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399
This theorem is referenced by:  rexeqi  2608  rexeqdv  2610  rexeqbi1dv  2612  unieq  3715  bnd2  4067  exss  4119  qseq1  6445  finexdc  6764  supeq1  6841  isomni  6976  ismkv  6995  sup3exmid  8683  exmidunben  11866  neifval  12236  cnprcl2k  12302  bj-nn0sucALT  13103  strcoll2  13108  sscoll2  13113
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