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Theorem rexeqdv 2680
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 2674 . 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 105    = wceq 1353   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461
This theorem is referenced by:  rexeqbidv  2686  rexeqbidva  2688  fnunirn  5771  cbvexfo  5790  fival  6972  nninfwlpoimlemg  7176  nninfwlpoimlemginf  7177  nninfwlpoim  7179  genipv  7511  exfzdc  10243  zproddc  11590  infssuzex  11953  nninfdcex  11957  ennnfonelemrnh  12420  grppropd  12899  dvdsrpropdg  13322  cnpfval  13835
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