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Theorem rexeqbi1dv 2670
Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
Hypothesis
Ref Expression
raleqd.1  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexeqbi1dv  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ps ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rexeqbi1dv
StepHypRef Expression
1 rexeq 2662 . 2  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
2 raleqd.1 . . 3  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
32rexbidv 2467 . 2  |-  ( A  =  B  ->  ( E. x  e.  B  ph  <->  E. x  e.  B  ps ) )
41, 3bitrd 187 1  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1343   E.wrex 2445
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450
This theorem is referenced by:  reg2exmid  4513  reg3exmid  4557  exmidomni  7106  bj-nn0suc0  13842
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