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Theorem reueq1 2906
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 2572 . 2  |-  F/_ x A
2 nfcv 2572 . 2  |-  F/_ x B
31, 2reueq1f 2902 1  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   E!wreu 2707
This theorem is referenced by:  reueqd  2914  isplig  21765  isfrgra  28380  frgra3v  28392  1vwmgra  28393  3vfriswmgra  28395  hdmap14lem4a  32672  hdmap14lem15  32683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-cleq 2429  df-clel 2432  df-nfc 2561  df-reu 2712
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