MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvreuv Structured version   Unicode version

Theorem cbvreuv 2934
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 1629 . 2  |-  F/ y
ph
2 nfv 1629 . 2  |-  F/ x ps
3 cbvralv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvreu 2930 1  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E!wreu 2707
This theorem is referenced by:  reu8  3130  aceq1  7998  aceq2  8000  fin23lem27  8208  divalglem10  12922  lspsneu  16195  lshpsmreu  29907
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-reu 2712
  Copyright terms: Public domain W3C validator