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Theorem reueqd 2858
Description: Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
Hypothesis
Ref Expression
raleqd.1  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
reueqd  |-  ( 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 reueqd
StepHypRef Expression
1 reueq1 2850 . 2  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
2 raleqd.1 . . 3  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
32reubidv 2836 . 2  |-  ( A  =  B  ->  ( E! x  e.  B  ph  <->  E! x  e.  B  ps ) )
41, 3bitrd 245 1  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   E!wreu 2652
This theorem is referenced by:  aceq1  7932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-cleq 2381  df-clel 2384  df-nfc 2513  df-reu 2657
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