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Theorem cbvrex 3078
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvral.1  |-  F/ y
ph
cbvral.2  |-  F/ x ps
cbvral.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrex  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvrex
StepHypRef Expression
1 nfcv 2616 . 2  |-  F/_ x A
2 nfcv 2616 . 2  |-  F/_ y A
3 cbvral.1 . 2  |-  F/ y
ph
4 cbvral.2 . 2  |-  F/ x ps
5 cbvral.3 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
61, 2, 3, 4, 5cbvrexf 3076 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   F/wnf 1621   E.wrex 2805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810
This theorem is referenced by:  cbvrmo  3080  cbvrexv  3082  cbvrexsv  3093  cbviun  4352  isarep1  5649  elabrex  6130  onminex  6615  boxcutc  7505  indexfi  7820  wdom2d  7998  hsmexlem2  8798  iundisj  22127  mbfsup  22240  iundisjf  27662  iundisjfi  27838  voliune  28441  volfiniune  28442  cvmcov  28975  indexa  30467  rexrabdioph  30970  rexfrabdioph  30971  elabrexg  31673  fprodle  31846  stoweidlem31  32055  stoweidlem59  32083  rexsb  32415  cbvrex2  32420  bnj1542  34335
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