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Theorem cbvrex 2761
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 2419 . 2  |-  F/_ x A
2 nfcv 2419 . 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 2759 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531   E.wrex 2544
This theorem is referenced by:  cbvrmo  2763  cbvrexv  2765  cbvrexsv  2776  cbviun  3939  onminex  4598  isarep1  5331  elabrex  5765  boxcutc  6859  indexfi  7163  wdom2d  7294  hsmexlem2  8053  iundisj  18905  mbfsup  19019  iundisjfi  23363  iundisjf  23365  cvmcov  23794  indexa  26412  rexrabdioph  26875  rexfrabdioph  26876  stoweidlem31  27780  stoweidlem59  27808  rexsb  27946  cbvrex2  27951  bnj1542  28889
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-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549
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