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Theorem rexeqbidv 2685
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
Hypotheses
Ref Expression
raleqbidv.1  |-  ( ph  ->  A  =  B )
raleqbidv.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rexeqbidv  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem rexeqbidv
StepHypRef Expression
1 raleqbidv.1 . . 3  |-  ( ph  ->  A  =  B )
21rexeqdv 2679 . 2  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ps )
)
3 raleqbidv.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
43rexbidv 2478 . 2  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. x  e.  B  ch )
)
52, 4bitrd 188 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)
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:  supeq123d  6984  ismnddef  12708  mndpropd  12730  mnd1  12734  isgrp  12770  isgrpd2e  12783  grp1  12862  issrgid  12987  isringid  13031  reldvdsrsrg  13083  dvdsrvald  13084
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