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Theorem reueq1 3025
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 2616 . 2  |-  F/_ x A
2 nfcv 2616 . 2  |-  F/_ x B
31, 2reueq1f 3021 1  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   E!wreu 2801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-eu 2266  df-cleq 2446  df-clel 2449  df-nfc 2604  df-reu 2806
This theorem is referenced by:  reueqd  3033  lubfval  15268  glbfval  15281  isplig  23817  isfrgra  30731  frgra3v  30743  1vwmgra  30744  3vfriswmgra  30746  hdmap14lem4a  35858  hdmap14lem15  35869
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