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Theorem cbvreuv 2766
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 1605 . 2  |-  F/ y
ph
2 nfv 1605 . 2  |-  F/ x ps
3 cbvralv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvreu 2762 1  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E!wreu 2545
This theorem is referenced by:  reu8  2961  aceq1  7744  aceq2  7746  fin23lem27  7954  divalglem10  12601  lspsneu  15876  lshpsmreu  29299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-cleq 2276  df-clel 2279  df-reu 2550
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