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Theorem reueq1 3000
Description: Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
reueq1  |-  ( 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 reueq1
StepHypRef Expression
1 nfcv 2602 . 2  |-  F/_ x A
2 nfcv 2602 . 2  |-  F/_ x B
31, 2reueq1f 2996 1  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1454   E!wreu 2750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-eu 2313  df-cleq 2454  df-clel 2457  df-nfc 2591  df-reu 2755
This theorem is referenced by:  reueqd  3008  lubfval  16272  glbfval  16285  isfrgra  25766  frgra3v  25778  1vwmgra  25779  3vfriswmgra  25781  isplig  25957  hdmap14lem4a  35486  hdmap14lem15  35497  uspgredg2vlem  39349  uspgredg2v  39350
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