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Theorem cbvrexv 2551
Description: Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
Hypothesis
Ref Expression
cbvralv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrexv  |-  ( 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 cbvrexv
StepHypRef Expression
1 nfv 1437 . 2  |-  F/ y
ph
2 nfv 1437 . 2  |-  F/ x ps
3 cbvralv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvrex 2547 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329
This theorem is referenced by:  cbvrex2v  2559  reu7  2758  reusv3  4219  funcnvuni  4995  fun11iun  5174  fvelimab  5256  fliftfun  5463  grpridd  5724  frecsuc  6021  nnaordex  6130  supmoti  6398  cardval3ex  6422  prarloclemlo  6649  prarloclem3  6652  cauappcvgprlemdisj  6806  cauappcvgprlemladdru  6811  cauappcvgprlemladdrl  6812  cauappcvgpr  6817  caucvgprlemdisj  6829  caucvgprlemcl  6831  caucvgprlemladdfu  6832  caucvgprlemladdrl  6833  caucvgpr  6837  caucvgprprlemell  6840  caucvgprprlemelu  6841  caucvgprprlemlol  6853  caucvgprprlemclphr  6860  caucvgprprlemexbt  6861  nntopi  7025  axcaucvglemres  7030  ublbneg  8644  qbtwnzlemstep  9204  qbtwnzlemshrink  9205  rebtwn2zlemstep  9208  rebtwn2zlemshrink  9209  cvg1nlemres  9811  resqrexlemoverl  9847  resqrexlemsqa  9850  resqrexlemex  9851  odd2np1lem  10182  bj-nn0sucALT  10469
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