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Theorem cbvreuv 3031
Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
cbvralv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvreuv  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Distinct variable groups:    x, A    y, A    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem cbvreuv
StepHypRef Expression
1 nfv 1674 . 2  |-  F/ y
ph
2 nfv 1674 . 2  |-  F/ x ps
3 cbvralv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvreu 3027 1  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   E!wreu 2794
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 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-cleq 2442  df-clel 2445  df-reu 2799
This theorem is referenced by:  reu8  3238  aceq1  8374  aceq2  8376  fin23lem27  8584  divalglem10  13694  lspsneu  17296  lshpsmreu  33036
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