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Theorem rexeqdv 2610
Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
Hypothesis
Ref Expression
raleq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
rexeqdv  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ps )
)
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rexeqdv
StepHypRef Expression
1 raleq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 rexeq 2604 . 2  |-  ( A  =  B  ->  ( E. x  e.  A  ps 
<->  E. x  e.  B  ps ) )
31, 2syl 14 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ps )
)
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:  rexeqbidv  2616  rexeqbidva  2618  fnunirn  5636  cbvexfo  5655  fival  6826  genipv  7285  exfzdc  9985  infssuzex  11569  ennnfonelemrnh  11856  cnpfval  12291
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